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6 

NEWCOMB'S 

Mathematical  Course. 

» 

I.    SCHOOL  COURSE. 

Algebra  for  Schools, $1.20 

Key  to  Algebra  for  Schools,       ....  1.20 

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HE/\fRY  HOLT  d  CO..  Publishers,  New  York. 


STBWCOifB'S   MATnEMATIOAL    COURSE 


ELEMENTS 


OP  THB 


DIFFERENTIAL    AND    INTEGRAL 


CALCULUS 


BY 


SIMOI^    ISTEWOOMB 

Ptofem>r  of  Mathematics  in  the  Johns  Hopkins  University 


NEW  YORK 
HENRY  HOLT  AND  COMPANY 

1887 


AS 


€•7 


A 


COPYMOHT,  1887, 
BY 

HENRY  HOLT  &  CO. 


use; 


iU 


PREFACE. 


The  present  work  is  intended  to  contain  about  as  much 
of  the  Calcuhis  as  an  undergraduate  student,  either  in  Arts 
or  Science,  can  be  expected  to  master  during  his  regular 
course.  He  may  find  more  exercises  than  he  has  time  to 
work  out;  in  this  case  it  is  suggested  that  he  only  work 
enough  to  show  that  he  understands  the  principles  they  are 
designed  to  elucidate. 

The  most  difficult  question  which  arises  in  treating  the 
subject  is  how  the  first  principles  should  be  presented  to  the 
mind  of  the  beginner.  The  author  has  deemed  it  best  to  be- 
gin by  laying  down  the  logical  basis  on  which  the  whole 
superstructure  must  ultimately  rest.  It  is  now  well  under- 
stood that  the  method  of  limits  forms  the  only  rigorous  basis 
for  the  infinitesimal  calculus,  and  that  infinitesimals  can  be 
used  with  logical  rigor  only  when  based  on  this  method,  that 
is,  when  considered  as  quantities  approaching  zero  as  their 
limit.  When  thus  defined,  no  logical  difficulty  arises  in  their 
use;  they  flow  naturally  from  the  conception  of  limits,  and 
they  are  therefore  introduced  at  an  early  stage  in  the  present 
work. 

The  fundamental  principles  on  which  the  use  of  infinitesi- 
mals is  based  are  laid  down  in  the  second  chapter.  But  it  is 
not  to  be  expected  that  a  beginner  will  fully  grasp  these  prin- 
ciples un'il  he  has  become  familiar  with  the  mechanical  pro- 
cess of  differentiation,  and  with  the  application  of  the  calcu- 


IV 


PREFACE. 


lus  to  special  problems.  It  may  therefore  be  found  best  to 
begin  with  a  single  careful  reading  of  the  chapter,  and  after- 
ward to  use  it  for  reference  a^  the  student  finds  occasion  to 
apply  the  principles  laid  down  in  it. 

The  author  is  indebted  to  several  friends  for  advice  and 
assistance  in  the  final  revision  of  the  work.  Professor  John 
E.  Clark  of  the  Sheffield  Scientific  School  and  Dr.  Fabian 
Franklin  of  the  Johns  Hopkins  University  supplied  sugges- 
tions and  criticisms  which  proved  very  helpful  in  putting  the 
first  three  chapters  into  shape.  Miss  E.  P.  Brown  of  Wash- 
ington has  read  all  the  proofs,  solving  most  of  the  problems  as 
she  went  along  in  order  to  test  their  suitability. 


nd  best  to 
and  after- 
occasion  to 


idvice  and 
issor  John 
)r.  Fabian 
3d  sugges- 
utting  the 
I  of  Wash- 
roblems  as 


CONTENTS. 


PART  I. 


THE  DIFFERENTIAL  0/LGULU8. 

Chapter  I.    Of  "V  abiables  and  Functions 

^1.  Nature  of  Functions.  2.  Their  Classification.  3.  anc- 
tional  Notation.  4.  Functions  of  Severa'  Variables.  5.  Func- 
tions of  Functions.  6.  Product  of  the  First  n  Numbers.  7.  Pi- 
nomial  CoeflScients.  8.  Graphic  Representation  of  Functions. 
9.  Continuity  and  Discontiauity  of  Functions.  10.  Many  valued 
Functions. 


PAGE 

3 


•j^ 


Chapter  II.    Of  Limits  and  Infinitesimals 17 

§  11.  Limits.  12.  Infinites  and  Infinitesimals.  13.  Properties. 
14.  Orders  of  Infinitesimals.    16.  Orders  of  Infinites. 

Chapter  III.    Of  Differentials  and  Derfvativrs 25 

§16.  Increments  of  Variables.  17.  First  Idea  of  Differentials 
and  Derivatives.  18.  Illustrations.  19.  Illustration  by  Velocities. 
20.  Geom  trical  Illustration. 

Chapter  IV.    Dk::perentiation  of  Explicit  Functions 31 

§  21.  The  P  "ocess  of  Dilierentiation  in  General.  23.  Differen- 
tials of  Sums.  23.  Differential  of  a  Multiple.  24.  Differential  of 
a  Constant.  25.  Differentials  of  Products  and  Powers.  26.  Dif- 
ferential of  a  Quotient  of  Two  Variables.  27.  Differentials  of  Ir- 
rational Expressions.  28.  Logarithmic  Functions.  ?9.  Expo- 
nential Functions.  30.  The  Trigonometric  Functions.  31.  Cir- 
cular Functions.  32.  Logarithmic  Differentiation.  33.  Velocity 
or  Derivative  with  Respect  to  the  Time. 

Chapter  V.    Functions  of  Several  Variables  and  Impli- 
cit Functions 54 

§  34.  Partial  Differentials  and  Derivatives.  35.  Total  Differen- 
tials.   36.  Principles  involved  in  Partial  Differentiation.    37.  Dif- 


▼I 


CONTENTS. 


fcrentiation  of  Implicit  Funciious.  88.  Implicit  Functions  of  Sev- 
eral Variables.  89.  Case  of  Implicit  Functions  expressed  by 
Simultaneous  Equations.  40.  Functions  of  Functions.  41.  Func- 
tions of  Variables,  some  of  which  are  Functions  of  tlie  Others. 
42.  Extension  of  the  Principle.  48.  Nomenclature  of  Partial 
Derivatives.  44.  Dependence  of  the  Derivative  upon  the  Form 
of  the  Function. 


PASE 


Chapter  VI.    Derivatives  op  Higher  Orders 74 

^45.  Second  Derivatives.  46.  Derivatives  of  Any  Order. 
47.  Special  Forms  of  Derivatives  of  Circular  and  Exponential 
Functions.  48.  Successive  Derivatives  of  an  Implicit  Function. 
40.  Successive  Derivatives  of  a  Product.  50.  Successive  Deriva- 
tives with  Respect  to  Several  Equicrescent  Variables.  51.  Result 
of  Successive  Differentiations  independent  of  the  Order  of  the 
Differentiations.  52.  Notation  for  Powers  of  a  Differential  or 
Derivative. 

Chapter  VII.    Special  Cases  op  Successive  Derivatives.  . .    86 
§  53.  Successive  Derivatives  of  a  Power  of  a  Derivative.  54.  De- 
rivatives of  Functions  of  Functions.    55.  Change  of  the  Equicres- 
cent Variable.    56.  Two  Variables  connected  by  a  Third. 

Chapter  VIII.    Developments  in  Series 95 

§  57.  Classification  of  Series.  58.  Convergence  and  Divergence 
of  Series.  59.  Maclaurin's  Theorem.  60.  Ratio  of  the  Circum- 
ference of  a  Circle  to  its  Diameter,  61.  Use  of  Symbolic  Nota- 
tion for  Derivatives.  62.  Taylor's  Theorem.  63.  Identity  of 
Taylor's  and  Maclaurin's  Theorems.  64.  Cases  of  Failure  of 
Taylor's  and  Maclaurin's  Theorems.  65.  Extension  of  Taylor's 
Theorem  to  Functions  of  Several  Variables.  66.  Hyperbolic 
Functions. 


Chapter  IX.    Maxima  and  Minima  op  Functions  op  a  Sin- 
gle Variable 117 

§67.  Definition  of  Maximum  Value  and  Minimum  Value. 
68.  Method  of  finding  Maximum  and  Minimum  Values  of  a  Func- 
tion. 69.  Case  when  the  Function  which  is  to  be  a  Maximum  or 
Minimum  is  expressed  as  a  Function  of  Two  or  More  Varir.bles 
connected  by  Equations  of  Condition. 


Chapter  X.    Indeterminate  Forms 128 

§70.  Examples  of  Indeterminate  Forms.    71.    Evaluation  of 

73.    Form  co  —  oo. 

00 

75.  Forms  0°  and  00  °. 


the  Form— .    72.  Forms  —  and  0  X  oo, 
0  00 


CONTENTS. 


VU 


Vkom 
CnAPTER  XI.    Op  Plane  Curves 187 

^  76.  Forms  of  the  Eiiuiitions  c.  Curves.  77.  Indnitcsimal  Ele- 
ments of  Curves.  78.  Properties  of  Intinitesimul  Arcs  and 
Chords.  70.  Expressions  for  Elements  of  Curves.  80.  Equa- 
tions of  Certain  Noteworthy  Curves.  The  Cycloid.  81.  The 
Lemniscate.  83.  The  Archimedean  Spiral.  88.  The  Logarith- 
mic Spiral. 

Chapter  XII.    Tangents  and  Normals 147 

^84.  Tangent  and  Normal  compared  with  Subtangent  and 
Subnormal.  85.  General  Equation  for  a  Tangent.  86.  Sub- 
tangent  and  Subnormal.  87.  Moditied  Forms  of  the  Equation. 
88.  Tangents  and  Normals  to  the  Conic  Sections.  80.  Length  of 
the  Perpendicular  from  the  Origin  upon  a  Tangent  or  Normal. 
00.  Tangent  and  Normal  in  Polar  Co-ordinates.  01.  Perpendicidar 
from  the  Pole  ujion  the  Tangent  or  Normal.  03.  Equation  of 
Tangent  and  Normal  derived  from  Polar  Equation  of  the  Curve. 

Chapter  XIII.    Op    Asymptotes,     Singular     Points     and 

Curve-tracing 157 

§03.  Asymptotes.  04.  Examples  of  Asympt'^tes.  05,  Points 
of  Inflection.  06.  Singular  Points  of  Curves.  07.  Condition  of 
Singular  Points.  08.  Examples  of  Double  points.  00.  Curve- 
tracing. 


Chapter  XIV.    Tiieory  op  Envelopes 169 

§100.  Envelope  of  a  Family  of  Lines.  101.  All  Lines  of  a 
Family  tangent  to  the  Envelope.   103.  Examples  and  Applications. 

Chapter  XV.  Op  Curvature,  Evolutes  and  Tnvolui'es 180 

§  103.  Position;  Direction;  Curvature.  104.  Contacts  of  DiflFer- 
ent  Orders.  105.  Intersection  or  Non-intersection  of  Curves  ac- 
cording to  the  Order  of  Contact.  106.  Radius  of  Curvature. 
107.  The  Osculating  Circle.  108.  Radius  ol  Curvature  when  the 
Abscissa  is  not  taken  as  the  Independent  Variable.  100.  Ra- 
dius of  Curvature  of  a  Curve  referred  to  Polar  Co-ordinates. 
110.  Evolutes  and  Involutes.    111.  Case  of  an  Auxiliary  Variable. 

113.  The  Evolutc  of  the  Parabola.     113.  E volute  of  the  Ellipse. 

114.  Evolute  of  the  Cycloid.    115.  Fundamental  Properties  of 
the  Evolute.     110.  Involutes. 


VIU 


CONTENTS. 


PART   II. 
THE  INTEGRAL  CALCULUS. 

PAOB 

CiiAPTEB  I.    The  Elementary  Forms  op  Integration 201 

^117.  Deflnition  of  Integrutiou.  118.  Arbitrary  Constant  of 
InUjgratlon.  110.  Integration  of  Entire  Functions.  120.  Tiie 
Logaritliinic  Function.  121.  Another  Metliod  of  obtaining  the 
Logarithmic  Integral.  122.  Exponential  Functions.  128.  The 
Elementary  Forms  of  Integration. 


Chapter  II.    Inteqrals    immediately    reducible    to    the 
Elementary  Forms 209 

§  124.  Integrals  reducible  to  the  Form  /  y^dy.  125.  Appli- 
cation to  the  Case  of  a  Falling  Body.  126.  lieduction  to  the  Loga- 
rithmic Form.     127.  Trigonometric  Forms.    128.  Integration  of 

and  ~ -::.    129.  Integrals  of  the  Form    /  - 

131. 


a"  4"  **  «*  —  «■■''  •         o  J  a-\-bx-\-eji^^' 

130.  Inverse  Sines  and  Cosines  as  Integrals.     131.  Two  Forms  of 
Integrals  expressed  by  Circular  Functions.    132.  Integration  of 

183.    Integration  of  -^        ^ ;;■    134.  Exponen- 


t/a'^ T  x^ 
tial  Forms. 

Chapter  III. 


^a-\-bx  ±  Gxi^ 


Integration  by  Rational  Transformations.  .  222 

x^dx  ,  xdx 


§185.  Integration  of  ^-^^^-dir, 


and 


««      "''{a-\-bx)»  a-\-bx±cx'*' 

136.  Reduction  of  Rational  Fractions  in  general.    137.  Integra- 
tion by  Parts. 


Chapter  IV.    Integration  op  Irrational  Algebraic  Dip- 

perentials 233 

§138.  When  Fractional  Powers  of  the  Independent  Variable 
enter  into  the  Expression.  139.  Cases  when  the  Given  Differen- 
tial Contains  an  Irrational  Quantity  of  the  Form   i^a  -{-  bx  -{-  cx^. 

dv 
140.  Integration  of  dO  =  —  .    141.  General  Theory 

r  yar"^  -\-hr  -  \ 

of  Irrational  Binomial  Differentials.  142.  Special  Cases  when 
m  -f- 1  =:  w,  or  m-\-\  -\-  np  =  —  n.  143.  Forms  of  Reduction 
of  Irrational  Binomials.  144.  Formulae  A  and  B,  in  which  m  is 
increased  or  diminished  by  n.  145.  Formulae  C  and  D,  in  which 
p  is  increased  or  diminished  by  1.  146.  Effect  of  the  Formulae. 
147.  Case  of  Failure  in  this  Reduction. 


CONTENTS. 


PAOK 


Chapter  V.    Integration  of  Transcendent  Functions 246 

^  148.  Integration  of     /  «"•*  cos  nxdx  and    /  «m*  sin  nxdx. 

149.  Integration  of  sin'"  a;  co8'»  xdx.    150.  Special  Cases  of  /  sin"*  x 

(Ix 

cos^xdx.   151.  Integration  of ——T-;; , — j — ~.     152.  Integra- 

*  m'  sm^  X  -f-  ii'  cos^  x  " 

dy 


tion  of  — I — i 


in'  sin"''  X  -f-  n'  cos"  x' 
153.  Special  Cases  of  the  Last  Two  Forms. 


a  +  *  cos  //■ 

154.  Integration  of  sin  mx  cos  nxdx.     155.  Integration  by  Devel- 
opment in  Series. 

Chapter  VI.  Of  Definite  Inteoralb 265 

^156.  Successive  Increments  of  a  Variable.  157.  Differential 
of  an  Area.  158.  The  Formation  of  a  Definite  Integial.  150.  Two 
Conceptions  of  a  Definite  Integral.  100.  Differentiation  of  a 
Definite  Integral  witli  respect  to  its  Limits.  161.  Examples  and 
Exerci.ses  in  finding  Definite  Integrals.  162.  Failure  of  the 
Method  when  the  Function  becomes  Infinite.  163.  Clmnge  of 
Variable  in  Definite  Integrals.  164.  Subdivision  of  a  Definite  In- 
tegral.   165.  Definite  Integrals  through  Integration  by  Parts. 

Chapter  VII.    Successivb  Integration 272 

§  166.  Differentiation  under  the  Sign  of  Integration.  167.  Ap- 
plication of  the  Principle  to  Definite  Integrals.  168.  Integration 
by  means  of  Differentiating  Known  Integrals.  169.  Application 
to  a  Special  Case.  170.  Double  Integrals.  171.  Value  of  a  Func- 
tion of  Two  Variables  obtained  from  its  Second  Derivative. 
173.  Triple  and  Multiple  Integrals.  173.  Definite  Double  Inte- 
grals. 174.  Definite  Triple  and  Multiple  Integrals.  175.  Product  of 


Chapter  VIII.    Rectification  and  Quadrature 

§177.  The  Rectification  of  Curves.  178.  The  Parabola.  179.  The 
Ellipse.  180.  The  Cycloid.  181.  The  Archimedean  Spiral. 
182.  The  Logarithmic  Spiral.  183.  The  Quadrature  of  Plane 
Figures.  184.  The  Parabola.  185.  The  Circle  and  the  Ellipse. 
18G.  The  Hyperbola.    187.  The  Lemniscate.     188.  The  Cycloid. 

Chapter  IX.    The  Cubature  op  VoiiUMES 

§189.  General  Formula).  190.  The  Sphere.  191.  The  Pyra- 
mid. 192.  The  Ellipsoid.  193.  Volume  of  any  Solid  of  Revolu- 
tion. 194.  The  Paraboloid  of  Revolution.  195.  The  Volume  gen- 
erated by  the  Revolution  of  a  Cycloid  around  its  Base.  196.  The 
Hyperboloid  of  Revolution  of  Two  Nai)pes.  197.  Ring-shaped 
Solids  of  Revolution.  198.  Application  to  the  Circular  Ring. 
199.  Quadrature  of  Surfaces  of  Revolution.  200.  Examples  of 
Surfaces  of  Revolution. 


285 


297 


:•! 


PART  I. 


THE  DIFFERENTIAL  CALCULUS. 


I  ill 


USE    OF   THE    SYMBOL  = 

The  symbol  =  of  identity  as  employed  in  this  work  indi- 
cates that  the  single  letter  on  one  side  of  it  is  used  to  repre- 
sent the  expression  or  thing  defined  on  the  other  side  of  it. 

When  the  single  letter  precedes  the  symbol  =,  the  latter 
may  commonly  be  read  is  put  for,  or  is  defined  as. 

Wnen  the  single  letter  follows  the  symbol,  the  latter  may 
be  read  which  let  ns  call. 

In  each  case  the  equality  of  the  quantities  on  each  side  of 
E  does  not  follow  from  anything  that  precedes,  but  is  assumed 
at  the  moment.  But  having  once  made  this  assumption,  any 
equations  which  may  flow  from  it  are  expressed  by  the  sign 
=,  as  usual. 


I 


PART  I. 
THE  DIFFERENTIAL  CALCULUS. 


s  work  indi- 

ed  to  repre- 

side  of  it. 

E,  the  latter 

s, 

3  latter  may 

sach  side  of 

is  assumed 

tnption,  any 

3y  the  sign 


% 


CHAPTER  I. 

OF  VARIABLES  AND  FUNCTIONS. 


1.  In  the  higher  mathematics  we  conceive  ourselves  to  be 
dealing  Avith  pairs  of  quantities  so  related  that  the  value  of 
one  depends  upon  that  of  the  other.  For  each  value  which 
wo  assign  to  one  we  conceive  that  there  is  a  corresponding 
value  of  the  other. 

For  example,  the  time  required  to  perform  a  journey  is  a 
function  of  the  distance  to  be  travelled,  because,  other  things 
being  equal,  the  time  varies  when  the  distance  varies. 

We  study  the  relation  between  two  such  quantities  by  as- 
signing values  at  pleasure  to  one,  r.nd  ascertaining  and  com- 
paring the  corresponding  values  of  the  other. 

The  quantity  to  which  we  assign  values  at  pleasure  is  called 
the  independent  variable. 

The  quantity  whose  values  depend  upon  those  of  the  inde- 
pendent variable  is  called  a  function  of  that  variable. 

Example  I.  If  a  train  travels  at  the  rate  of  30  miles  an 
I  hour,  and  if  we  ask  hoAv  long  it  will  take  the  train  to  travel 
15  miles,  30  miles,  60  miles,  900  miles,  etc.,  we  shall  have  for 
the  corresponding  times,  or  functions  of  the  distances,  half  an 
hour,  one  hour,  two  hours,  thirty  liours,  etc. 


THE  DIFFERENTIAL  CALCULUS. 


I 


In  thinking  thus  we  consider  the  distance  to  be  travelled  as 
the  independent  variable,  and  the  time  as  the  function  of  the 
distance. 

Example  II.  If  between  the  quantities  x  and  y  wo  have 
the  equation 

y  =  2ax^f 

we  may  suppose 

a;  =  -  1,  0,  +  1,  +  2,  +  3,  etc., 
and  we  shaU  then  have 

y  =  2a,  0,  2a,  Sa,  18a,  etc. 

Here  x  is  taken  as  the  independent  variable,  and  y  as  the 
function  of  x.  For  each  value  we  assign  to  x  there  is  a  corre- 
sponding value  of  y. 

When  the  relation  between  the  two  quantities  is  expressed 
by  means  of  an  equation  between  symbolic  expressions,  the 
one  is  called  an  analytic  function  of  the  other. 

An  anal}i;ic  function  is  said  to  be 

Explicit  when  the  symbol  which  represents  it  stands 
alone  on  one  side  of  the  equation; 

Implicit  when  it  does  not  so  stand  alone. 

Example.  In  the  above  equation  y  is  an  explicit  function 
of  X.     But  if  we  have  the  equation 

y'  +  ^y  =  x\ 

then  for  each  value  of  x  there  will  be  a  certain  value  of  y, 
which  will  be  found  by  solving  the  equation,  considering  y  as 
the  unknown  quantity.  Here  y  is  still  a  function  of  x,  be- 
cause to  each  value  of  x  corresponds  a  certain  value  of  y,  but 
because  y  does  not  stand  alone  on  one  side  of  the  equation  it 
is  called  an  implicit  function. 

Kemark.  The  difference  between  explicit  and  implicit 
functions  is  merely  one  of  form,  arising  from  the  different 
ways  in  which  the  relation  may  be  expressed.  Thus  in  the 
two  forms 


VARIABLES  AND  FUNCTIONS. 


be  travelled  as 
1  notion  of  the 

,nd  y  wo  have 


and  y  as  the 
liere  is  a  corre- 

is  is  expressed 
tpressions,  the 
er. 

jnts   it  stands 


Dlicit  function 


n  value  of  //, 
nsidering  y  as 
tion  of  X,  be- 
ilue  of  y'y  but 
le  equation  it 

and  implicit 
the  different 
Thus  in  the 


y  =  2aa;', 

y  —  2ax^  =  0, 

\y  is  the  same  function  of  x]  but  its  form  is  explicit  in  the  firs^-, 
[and  implicit  in  the  second. 

An  implicit  function  may  be  reduced  to  an  explicit  one  by 
[solving  the  equation,  regarding  the  function  as  the  unknown 
[quantity.  But  as  the  solution  may  be  either  impracticable 
lor  too  complicated  for  convenient  use,  it  may  be  impossible  to 
[express  the  function  otherwise  than  in  an  implicit  form. 

3.  Classification  of  Functions,  When  y  is  an  explicit 
Ifunttion  of  x  it  is,  by  definition,  equal  to  a  symbolic  expression 
[containing  the  symbol  x.     Hence  we  may  call  either  y  or  the 

|symbolic  expression  the  function  of  x,  the  two  being  equiva- 
i  lent.  Indeed  any  algebraic  expression  containing  a  symbol  is, 
|by  definition,  a  function  of  the  quantity  represented  by  the 
jeymbol,  because  its  value  must  depend  upon  that  of  the  sym- 
Ibol. 

Every  algebraic  expression  indicates  that  certain  operations 

fare  to  be  performed  upon  the  quantities  represented  by  the 

I  symbols.     These  operations  are: 

1.  Addition  and  subtraction,  included  algebraically  in  one 
lass. 

2.  Multiplication,  including  involution. 

3.  Division. 

4.  Evolution,  or  the  extraction  of  roots. 
A  function  which  involves  only  these  four  operations  is 

ailed  algebraic. 
Functions  are  classified  according  to  the  operations  which 
ust  be  performed  in  order  to  obtain  their  values  from  the 

alues  of  the  independent  variables  upon  which  they  depend. 
A  rational  function  is  one  in  which  the  only  operations 

ndicated  upon  or  with  the  independent  variable  are  those  of 

ddition,  multiplication,  or  division. 


6 


THE  DIFFERENTIAL  CALCULUS. 


\m 


An  entire  function  is  a  rational  one  in  which  the  only  in- 
dicated operations  are  those  of  addition  and  multiplication. 
Examples.    The  expression 

a-\-hx-\-  ex*  -\-  dx* 

is  an  entire  function  of  x,  as  well  as  of  a,  h,  c  and  d. 

The  expression 

.  m  .        c 

X      x*  -\-nx 

is  a  rational  function  of  x,  but  not  an  entire  function  of  x. 

An  irrational  function  of  a  variable  is  one  in  which  the 
extraction  of  some  root  of  an  expression  containing  that  vari- 
able is  indicated. 
Example.    The  expressions 

Va  -f  bxy    {a  +  mx*  -\-  nx*) 

are  irrational  functions  of  x. 

Functions  which  cannot  be  represented  by  any  finite  com- 
bination of  the  algebraic  operations  above  enumerated  are 
called  transcendental. 

An  exponential  function  is  one  in  which  the  variable 
enters  into  an  exponent. 

Example.    The  expressions 

{a  +  «;)"»,    a^ 

are  entire  functions  of  x  when  n  and  y  are  integers.  But 
they  are  exponential  functions  of  y. 

Othor  transcendental  functions  are; 

Trigonometric  functions,  the  sine,  cosine^  etc. 

IiOgarithmic  functions,  which  require  the  finding  of  a 
logarithm. 

CirculsLT  functions,  which  are  the  inverse  of  the  trigo- 
nometric functions;  for  example,  if 

y  =  a  trigonometric  function  of  x,  sin  x  for  instance, 

then  :r  is  a  circular  function  of  y,  namely,  the  arc  of  which  y 
is  the  sine. 


VARIABLES  AND  FUN0TI0N8. 


h  the  variable 


integers.    But 


)  of  the  trigo- 


3.  Functional  Notation.  For  brevity  and  generality  we 
lay  represent  any  lunction  of  a  'variable  by  a  single  symbol 
laving  a  mark  to  indicate  the  variable  attached  to  it,  in  any 
form  we  may  elect.  Such  a  symbol  is  called  a  functional 
lymbol  or  a  symbol  of  operation. 

The  most  common  functional  symbols  are 

F,    f    and    0; 

)ut  any  signs  or  mode  of  writing  whatever  may  be  used. 
[Then,  such  expressions  as 

F{x),   f(x),    <p{x), 
{each  mean 

"  some  symbolic  expression  containing  s;." 

The  variable  is  enclosed  in  parentheses  in  order  that  the 
If  unction  may  not  be  mistaken  for  the  product  of  a  quantity 
\Fy  /  or  0  by  x. 

Identical  Functions.  Frnctions  which  indicate  identical 
[operations  upon  two  variables  are  considered  as  identical. 

Example.    If  we  consider  the  expression 

a  +  hy 
[as  a  certain  function  of  y,  then 

a-\-bx 
[is  that  same  function  of  x,  and 

a-\-l(x-{-  y) 
|is  that  same  function  ot  x-\-  y. 

When  the  functional  notation  is  applied,  then: 

Identical  functions  are  represented  by  the  same  functional 

^symbols. 

Examples.    If  we  put 

F{x)  =  a-^bx, 

fwe  shall  have  F{y)  =  a-^-by; 

W)  =  «  +  hfy 
F(x*-f)  =  a  +  b(x'-f), 


8 


THE  DIFFEBELTIAL  CALCULUS, 


;       i 


In  general,  If  we  define  afunctional  symbol  as  representing 
a  certain  function  of  a  variable,  that  same  symbol  applied  to 
a  secojid  variable  will  represent  J/ie  expression  formed  by  sub' 
stituting  the  second  variable  for  the  first. 

In  applying  this  rule  any  expression  may  be  regarded  as  a 
variable  to  be  substituted,  as,  in  the  last  example,  we  used 
X*  —  y*  as  a,  variable  to  be  substituted  for  x  in  the  original 
expression. 

EXERCISES. 

1.  If  we  put 

<p{x)  =  ax*f 

it  is  required  to  form  and  reduce  the  functions 

0(.V),     0(*),     0M,     0(-a;),     0(0;'),     0(|). 

2.  Putting 

it  is  required  to  form  and  reduce 

^(^+1),  ^(3-  ^©.  ^(f).  ^(i)+^{^). 

3.  Putting 

it  is  required  to  form  and  reduce 

fix -a),    Ax  +  a),    /(l).    /(I). 

4.  If  <p{x)  =  a*x  -\-  ex*, 
form  and  reduce  the  expressions 

0(a:'),     0(a'),     (^(ax),     (f>{bx),     <f){a -{- c),     <p{a  —  c). 

5.  Suppose  (f){x)  =  ax^  —  a^x,  and  thence  form 

0(.y),  0(2^),  0(*y), 

0(«  +  y),  <P{x  +  «),  0(«  -  «), 


IS 


VARIABLES  AND  FUNCTIONS.  0 

6.  Suppose /(a;)  =  a;',  and  thonce  form  the  values  of 

/(I),  /(^')>  A^')»  A^')>  A^'h  A^% 

7.  Let  us  put  0(m)  =  m(m  —  1)  (m  —  3)  (m  —  3);  thenco 
ind  the  values  of 

K6),  0(5),  0(4),  0(3),  0(2),  0(1),  0(0),  0(-  1),  0(-  2). 

8.  Prove  that  if  we  put  (f){x)  =  a*,  we  shall  have 
0(a:  +  y)  -  0(^)  X  0(y);        0(a:y)  =  [0(2:)]"  =  [0(2/)]'. 

4.  Functions  of  Several  Variables.  An  algebraic  expres- 
sion containing  several  quantities  may  be  represented  by  any 
Symbol  having  the  letters  which  represent  the  quantities  at- 
tached. 

ExAMPTES.    We  may  put 

0(a:,  y)^ax  —  ly, 

the  comma  being  inserted  between  x  and  ;/  so   that  their 
)roduct  shall  not  be  understood.     We  shall  then  have 

0(m,  n)  =  am  —  5w, 
0(y,  x)-ay  —  hx, 

jthe  letters  being  simply  interchanged; 

0(«  ■\-y,^-y)  =  ci{x-^y)-  l{x  -  y) 
=  (a  -  l)x  +  (a  +  h)y; 
<f>(a,  h)  =  a'-  h') 
<f){J),  a)  =  ab  —  ia  =  0; 
0(«  4-  b,  ab)  =  a(a  -\-b)  —  aJ'; 
0(fl!,  «)  =  «'  —  Ja; 


etc. 


etc. 


If  we  put  0(a,  b,  c)~2a-\-^b  —  5c,  we  shall  have 

0(a^*  «>  2/)  =  2^  +  3«  —  hy, 
p(z,  y,  x)  =  2z  +  3y  -  6x; 
0(w,  m,  —m)  =  2m  f  37?i  -f-  5m  =  10m; 
0(3,  8,  6)  =  3-3  -f  3-8  -  5*6  =  0. 


10 


THE  DIFFERENTIAL  CALCULUS. 


Let  ns  put 


\ 


3.  0(3,4). 

9.  /(7,  -  3). 
12.  /(^»,  rt,  2). 
15.  f(—a,—b,—ab). 


EXERCTGES. 

0(a:,  y)  EE  32;  -  4y; 
/(a;,  y)  E  ax'\-by; 
A^f  y»  z)  -,  ax  -f  Jy 
Thence  form  the  expressions: 
I.  0(y,  x),  2.  0(a,  3). 

4.  0(4,  3).  s.  0(10,  1). 

7.  /(J,  fl).  8.  /(y,  x), 

^^'  f{qy  -P)-         "•  /(^J.  a;,  y). 
13.  /(«.  ^  c).         14.  /(«',  *',  c')- 
Sometimes  there  is  no  need  of    any  functional  symbol 
except  the  parentheses.     For  example,  the  form  {m,  n)  may 
be  used  to  indicate  any  function  of  m  and  n, 

EXERCISES. 

Let  us  put    (m,  n)  ~  -^. —} ^^,  I 

^        '       n{n  —  1)  (w  —  2)  ' 

then  find  the  values  of — 

I.  (3,  3).  2.  (4,  3).  3.  (5,  3). 

4.  (6,  3).  5.  (7,  3).  6.  (8,  3). 

7.  (2,  -  1).  8.  (3,  -  2).  9.  (4,  -  2). 

6.  Functions  of  Functions,     By  the  definitions  of  the  pre- 
ceding chapter,  the  expression 

/(0(^)) 

will  mean  the  expression  obtained  by  substituting  (/){x)  for  x 
mf{x). 
We  may  here  omit  the  larger  parentheses  and  write  f(p{x) 

instead  of  /[  <p{x)  ] . 


For  example,  using  the  notation  of  exercises  1  and  3  of 

3,  we  shall  have 

-  ^  /  X      ax*  —  a      X*  —  1 

*'  iX  ~~  Cl\ 


VARIABLES  AND  FUNCTIONS, 


11 


s  1  and  3  of 


For  brevity  we  use  the  notation 

<t>\x)  =  0(0(3:)). 
Continuing  tho  same  system,  we  hare 

0*(^)  =  0(0'W)  =0'(0(^)); 

0*(a:)  =  0(0*(a;))=0'(0(a:)); 
eto.  etc.  etc. 

Examples.    1.  If 

(p{x)  =  ax*, 
then  0'(^)  =  «(«^')'  =  «V; 

0*(a;)  =  a{a*xy  =  a  V; 
etc.  etc.         etc. 


3.  If 

then 


/{x)  ~  a  —  Xy 
P{x)  =z  a  —  {a  —  x)  =  X) 

f\x)  =  a  —  f'ix)  —  a  —  x; 
and,  in  general. 

Remark.  The  functional  nomenclature  may  be  simplified 
to  any  extent. 

1.  The  parentheses  are  quite  unnecessary  when  there  is  no 
danger  of  mistaking  the  form  for  a  product. 

2.  When  it  is  once  known  what  the  variables  are,  we  may 
write  the  functional  symbol  without  them.  Thus  the  symbol 
0  may  be  taken  to  mean  fpx  or  <p{x). 

6.  Product  of  the  First  n  Numbers.  The  symbol  n\,  called 
factorial  n,  is  used  to  express  the  product  of  the  first  n  num- 
bers, 

1-3-3-...W. 

Thus,  ,  1!  =  1; 

2!  =  1-2  =  2; 
3!  =  l-2-3  =  6; 
4!  =  l-2'3-4  =  24; 

etc.  etc. 


12 


THE  DIFFERENTIAL  CALCULUS. 


!lf 


It  will  be  seen  that 

21  =  211; 

31  =  3-21; 
and,  in  general,         n\  =  w  (n  —  1)1, 

whatever  number  n  may  represent. 

EXERCISES. 

Compute  the  values  of — 

I.  51  2.  61  3.  81 

•7!  8! 

"*•  31  41  ^'  31  61 

6.  Prove  the  equation  2 •  4  •  6 •  8 • . .  .2m  =  2"^! 

7.  Prove  that,  when  n  is  even, 

Uy  _  n{n  —  2)  (n  —  4)..  .4-2 
-  ;;  . 

7.  Binomial  Coefficients.     The  binomial  coefficient 

n{n  —  1)  (n  —  2) ...  .to  5  terms 
1-2-3-...S 

is  expressed  in  the  abbreviated  form 

the  parentheses  being  used  to  distinguish  the  expression  from 


the  fraction  -. 

s 

EXAMPLES. 

• 

CO 

II 

CO  liH 
II 

CO  li-l 

/7\      7-6-5-4-3 
\5/  ~l-2-3-4-5~ 

(n\      n 

lrj=r  =  »- 

ln\  _  m(»  —  1)  (»  —  S) 

1-2-3 


VARIADLES  AND  FUNCTIONS. 


13 


EXERCISES. 


Prove  the  formula): 

'•  (I)  -  (a)- 

/6\_  J! 

ln  +  l\  _  n±l  (n\ 
5-  [^  ^  1;  -.v-|.  i  \sl- 


''   (2-)  =  fe)- 

'•  [1)  = 


if  I  (m  —  8)1 

-  (r) + (8 = m- 
'•(7)+(^)=m«-(i)+(i)=(^ 


8.  Graphic  Reprenentation  of  Functions.  The  methods  of 
Analytic  Geometry  enable  us  to  represent  functions  to  the  eye 
by  means  of  curves.  The  common  way  of  doing  this  is  to 
represent  the  independent  variable  by  the  abscissa  of  a  point, 
and  the  corresponding  value  of  the  function  by  its  ordinate. 
Let  x^y  .r„  0*3,  etc.,  be 
different  values  of  the  in- 
dependent variable,  and 
y.y  y,y  y,y  etc.,  the  cor- 
responding values  of  the 
function.  AVe  lay  ofE 
upon  the  axis  of  abscis- 
sas   the    lengths    0X„ 


1! 


y 


L 


?r^' 


y^ 


Pi.  Ps. 


Vz 


Vz 


Xi    X2    Xa 


Fio.  1. 


OX^,  OX^,  etc.,  equal 
to  rr„  x^,  a^3,  etc.,  and  terminating  at  the  points  X^,  X„  X„ 
etc.  At  each  of  these  points  we  erect  a  perpendicular  to  rep- 
resent the  corresponding  value  of  y.  The  ends,  /*,,  P„  P„ 
of  these  perpendiculars  will  generally  terminate  on  a  curve 
line,  the  form  of  which  shows  the  nature  of  the  function. 

It  must  be  clearly  seen  and  remembered  that  it  is  not  the 
curve  itself  which  represents  the  values  of  the  function,  but 
the  ordinates  of  the  curve. 


14 


THE  DIFFERENTIAL  CALCULUS. 


I 


i» 


T 


Fio.  2. 


9.  Continuitij'  and  Discontinuity  of  Functions.  Let  us 
consider  the  graphic  representation  of  a  function  in  the  most 
general  way.  We  measure  off  a  series  of  values,  OX^,  OX^, 
OX^,  etc.,  of  the  independent  variable,  and  at  the  points  Ji 
X,,  Xj,  etc.,  we  erect  ordinates. 
In  order  that  the  variable  ordinate 
may  actually  be  a  function  of  x  it  is 
sufficient  if,  for  every  value  of  the 
abscissa,  there  is  a  corresponding 
value  of  the  ordinate. 

Now  we  might  ct^nceive  of  such 
a  function  that  there  should  bo  no 
relation  between  the  different  val- 
ues of  the  ordinates,  but  that  every 
separate  point  should  have  its  own 
separate  ordinate,  as  shown  in 
Fig.  2.  If  this  remained  true  how 
numerous  soever  we  made  the  ordi- 
nates, then  the  ends  of  the  latter  would  not  terminate  in  any 
curve  at  all,  but  would  be  scattered  over  the  plane.  Such 
a  function  would  be  called  discontinuous  at  every  point. 

Such,  however,  is  not  the  kind  of  functions  commonly 
considered  in  mathematics.  The  functions  with  which  we 
are  now  concerned  are  such  that,  however  irregular  they  may 
appear  when  the  values  of  x  are  widely  separated,  the  ends 
of  the  ordinates  will  terminate  in  a  curve  when  we  bring 
those  values  close  enough  together. 

If  a  function  is  such  that  when  the  point  representing  the 
independent  variable  moves  continuously  from  X,  to  X^  (Fig. 
1)  the  end  of  the  ordinate  describes  an  unbroken  curve,  then 
we  call  the  function  continuous  between  the  values  x^  and 
x^  of  the  independent  variable. 

If  the  curve  remains  unbroken  how  far  soever  we  suppose 
X  to  increase,  positively  or  negatively,  we  call  the  function 
continuous  for  all  values  of  the  indejyendent  variable. 


VARIABLES  AND  FUNCTIONS. 


16 


But  if  there  is  a  value  a  oix  for  which  there  is  a  break  of 
any  kind  in  the  curve,  we  call  the  function  discontinuous  for 
the  value  a  of  the  independent  variable. 

Let  us,  for  example,  consider  the  function 


y- 


ar 


b{a  —  x)' 


Let  us  measure  off  on  the  axis  of  abscissas  the  length  OX 
=  a .  Then  as  we  make  our  varying  ordinate  approach  X 
from  the  left  it  will  increase  positively  without  limit,  and  the 
curve  will  extend  upwards  to  infinity;  if  we  approach  X  from 
the  right-hand  side,  the  ordinate  will  be  negative  and  the 
curve  will  go  downwards  to  infinity.  Thus  the  curve  will  not 
form  a  continuous  branch  from  the  one  side  to  the  other. 
Thus  the  above  function  is  discontinuous  for  the  value  a  of  x. 


o 


/ 


«^ 


a 


X 


Fig.  3. 


10.  Many-valued  Functions.  In  all  that  precedes,  we 
have  spoken  as  if  to  each  value  of  the  independent  variable 
corresponded  only  one  value  of  the  function.     But  it  may 


16 


THE  DIFFERENTIAL  CALCULUS. 


\\ 


happen  that  there  are  several  such  values.     For  example^  if  y 
is  au  implicit  function  of  x  represented  by  the  equation 

y*  +  fnxy*  -f  ^^^'y  •\-p^*  =  ^f 

then  we  know,  by  the  theory  of  equations,  that  there  will  be 
three  values  of  y  for  each  value  assigned  to  the  variable  x. 

Def.  According  as  a  function  admits  of  one,  two  or  n 
values,  it  is  called  one-valued,  two-valued  or  7i-valued. 

Infinitely-valued  Functions.  It  may  happen  that  to  each 
value  of  the  variable  there  are  an  infinity  of  different  values 
of  the  function.  A  case  of  this  is  the  function  sin  ^~  ^^  x,  or 
the  arc  of  which  x  is  the  sine.  This  arc  may  be  either  the 
smallest  arc  which  has  x  for  its  sine,  or  this  smallest  arc  in- 
creased by  any  entire  number  of  circumferences. 

Take,  for  example,  the  arc  whose  sine  shall 
be+i. 

The  two  smallest  arcs  will  be 

30°  =  \7r    and    150°  =  f  ;r. 

But  if  we  take  the  function  in  its  most  gen- 
eral sense  it  may  have  any  of  the  values 

(2+i);r;     (4  +  i)7r;     (6  +  i);r,     etc., 
or    (3-|-f)7r;     (4  +  |);r;     (G  +  f)^,     etc. 

When  we  represent  an  w-valued  function 
graphically,  there  will  be  n  values  to  each  ordi- 
nate. Hence  each  ordinate  will  cut  the  curve 
in  n  points,  real  or  imaginary. 

The  figure  in  the  margin  represents  the  infi- 
nitely-valued function 

y  =  a  sm  ^    '  -. 
^  a 

When  —  a  <x  <  -\-a,  any  ordinate  will  cut 
the  curve  in  an  infinity  of  points.  '%ia  *4 


1 

/     • 

/ 

1    y 

1/ 

1/ 

( 

\ 

\ 

^ 

V 

\ 

\ 

. 

\ 

/ 

/ 

/ 

1 

y 

1         y 

^ 

1    X 

l/ 

1 

l\ 

1  \ 

1    \ 

1  ^ 

\ 

1 

\ 

1 

1 

^ 

1 

J 
1 

) 

!    o 

/ 

X 

LIMITS  AND  INFINITESIMALS. 


17 


CHAPTER  II. 

OF  LIMITS  AND  INFINITESIMALS. 

11,  Limits.  The  method  of  limits  18  an  indirect  method 
of  arriving  at  the  values  of  certain  quantities  which  do  not 
admit  of  direct  determination.  The  method  rests  upon  the 
following  axioms  and  definition: 

Axiom  I.  Any  quantity,  however  small,  may  be  multiplied 
by  so  great  a  number  as  to  exceed  any  other  quantity  of  the 
same  kind,  however  great,  to  which  a  fixed  value  is  assigned. 

Axiom  II.  Conversely,  any  quantity,  however  great,  may 
be  divided  into  so  many  parts  that  each  part  shall  be  less  than 
any  other  quantity  of  the  same  kind,  however  small,  to  which 
a  fixed  value  is  assigned. 

Axiom  III.  Any  quantity  may  be  divided  into  any  num- 
ber of  parts ;  or  multiplied  any  number  of  times. 

Def.  The  limit  of  a  variable  quantity  X  is  a  quantity  L, 
which  we  conceive  Xto  approach  in  such  a  way  that  the  dif- 
ference L  —  X  becomes  less  than  any  quantity  we  can  name, 
but  which  we  do  not  conceive  X  to  reach. 

Example.  If  we  have  a  variable  quantity  X  and  a  con- 
stant quantity  L,  and  if  X,  in  varying  according  to  any  mathe- 
matical law,  takes  the  successive  values 

L  ±  0.1, 

L  ±  0.01, 

L  ±  0.001, 

L  ±  0.0001, 

and  so  on  indefinitely,  without  becoming  equal  to  L,  then  we 
say  that  L  is  the  limit  of  x. 


m 


I 


y 


n; 


I' 


ff 


I 


'I    i 


18 


THE  DIFFERENTIAL  CALCULUS. 


12,  Infinites  and  Infinitesimals.     Definitions. 

1.  An  infinite  quantity  is  one  considered  as  becoming 
greater  than  any  quantity  which  we  can  name. 

3.  An  infinitesimal  quantity  is  one  considered  in  the 
act  of  becoming  less  than  any  quantity  which  we  can  name; 
that  is,  in  the  act  of  approaching  zero  as  a  limit. 

3.  A  finite  quantity  is  one  which  is  neither  infinite  nor  in- 
finitesimal.* 

Examples.     If  of  a  quantity  x  we  either  suppose  or  prove 

X  >  10, 
X  >  100, 
X  >  100000, 

and  so  on  without  end,  then  x  is  called  an  infinite  quantity. 
If  of  a  quantity  h  we  either  suppose  or  prove 

h  <  0.1, 
h  <  0.001, 
h  <  0.00001, 

and  so  on  without  end,  then  h  is  an  infinitesimal  quantity. 

The  preceding  conceptions  of  limits,  infinites  and  infinitesi- 
mals are  applied  in  the  following  ways:  Let  us  have  an  inde- 
pendent variable  x,  and  a  function  of  that  variable  which  we 
call  y. 

Now,  in  order  to  apply  the  method  of  limits,  we  may  make 
three  suppositions  respecting  the  value  of  x,  namely: 

1.  That  X  approajhes  some  finite  limit. 

2.  That  X  increases  without  limit  (i.e.,  is  infinite). 

3.  That  X  diminishes  without  limit  (i.e.,  is  infinitesimal). 
In  each  of  these  cases  the  result  may  be  that  y  approaches 

a  finite  limit,  or  is  infinite,  or  is  infinitesimal. 


*  Strictly  speaking,  the  words  infinite  and  infinitesimal  are  both  adjec- 
tives qualifying  a  qvantity.  But  the  second  has  lately  been  used  also  as 
a  Doun,  and  we  shall  therefore  use  the  word  infinite  as  a  noun  meaning 
in  unite  quantity. 


LIMITS  AND  INFINITESIMALS. 


19 


Bred  in  the 
can  name; 


inite  nor  in- 


For  example,  let  us  have 

y 


X  -\-  a 


X  —  a 
Then— 
When  X  approaches  the  limit  a,  y  becomes  infinite. 
When  X  becomes  infinite,  y  approaches  the  limit  +  1- 
When  X  becomes  infinitesimal,  y  approaches  the  limit  —  1 . 

The  symbol  =,  followed  by  that  of  zero  or  a  finite  quantity, 
moans  *' approaches  the  limit."  The  symbols  ioo  mean 
^'increases  without  limit"  or  "becomes  infinite."  Hence 
the  three  last  statements  may  be  expressed  symbolically,  as 
follows: 

X  -\-  a 


When    X  =  a. 


When 


ic  i  CO, 
etc. 


then 


then 


X  —  a 

X  -{-  a 

X  —  a 

etc. 


00 


=  +  1; 


The  same  statements  are  more  commonly  expressed  thus: 

a) 


lim.  {x 


oo 


X  —  a 


lim.  ^-±^(a;ico)=  +1; 
X  —  a^  ' 

lim.  ?-±-^  (a;  i  0)    =  -  1. 
X  —  a^         ' 

13.  Properties  of  Infinite  and  Infinitesimal  Quantities. 

Theorem  I.  The  product  of  an  infinitesimal  ly  any  finite 
factor,  however  greats  is  an  infinitesimal. 

Proof.  Let  h  be  the  infinitesimal,  and  n  the  finite  factor 
by  which  it  is  multiplied.  I  say  how  great  soever  n  may  be, 
nh  is  also  an  infinitesimal.  For,  if  nh  does  not  become  less 
than  any  quantity  we  can  name,  let  or  be  a  quantity  less  than 
which  it  does  not  become.     Then  if  we  take,  as  we  may. 


we  shall  have 


n 
nh  <  a. 


(Axiom  III.) 


um 


:•'!  ■■ 


I! 


20 


TFE  DIFFERENTIAL  CALCULUS. 


That  is,  nh  is  less  than  a  and  not  less  than  a,  which  is 
absurd. 

Hence  nh  becomes  less  than  any  quantity  we  can  name, 
and  is  therefore  infinitesimal,  by  definition. 

Theorem  II.  The  quotient  of  an  infinite  qumitity  ly  any 
finite  divisor,  hotvever  great,  is  infinite. 

Proof.  Let  X  be  the  infinite  quantity,  and  n  the  finite 
divisor.  It  X  —  n  does  not  increase  beyond  every  limit,  let 
K  be  some  quantity  which  it  cannot  exceed.     Then  \jy  taking 


we  shall  have 


X>  nK, 


X      ^ 


(Ax.  III.) 


that  is,  —  greater  than  the  quantity  which  it  cannot  exceed, 

which  is  absurd. 

Hence  X—-n  increases  beyond  every  limit  we  can  name 
when  X  does,  and  is  therefore  infinite  when  X  is  infinite. 

Theorem  III.  TJie  product  of  any  finite  quantity,  how- 
ever small,  hy  an  infinite  rnultiplier,  is  infinite. 

This  follows  at  once  from  Axiom  I.,  since  by  increasing  the 
multiplier  we  may  make  the  product  greater  than  any  quan- 
tity we  can  name. 

Theorem  IV.  The  quotient  of  any  finite  quantity,  how- 
ever great,  by  an  infinite  divisor  is  i?ifinitesimaL 

This  follows  at  once  from  Axiom  II.,  since  by  increasing 
the  divisor  the  quotient  may  be  made  less  than  any  finite 
quantity. 

Theorem  V.  The  reciprocal  of  an  infinitesimal  is  an  in- 
finite, and  vice  versa. 

Let  h  be  an  infinitesimal.  If  j-  is  not  infinite,  there  must 
be  some  quantity  which  we  can  name  which  j  ^o^s  not  ex- 


LIMIT8  AND  INFINITESIMALS. 


21 


3  can  name, 


innot  exceed, 


nal  is  an  in- 


ceed.    Let  K  be  that  quantity.     Because  h  is  infinitesimal, 
we  may  have 


A<i 


which  gives 


h 


>^; 


that  is,  Y  greater  than  a  quantity  it  can  never  exceed,  which 

is  absurd. 
The  converse  theorem  may  be  proved  in  the  same  way. 

14.  Orders  of  Infinitesimals.  Def.  If  the  ratio  of  one 
infinitesimal  to  another  approaches  a  finite  limit,  they  are 
called  infinitesimals  of  the  ,samo  order. 

If  the  ratio  is  itself  infinitesimal,  the  lesser  infinitesimal  is 
said  to  be  of  higher  order  than  the  other. 

Theorem  VI.  If  we  have  a  series  proceeding  according 
to  the  powers  of  h, 

A+Bh-i-  Ch'  +  Dh'  +  etc., 

in  luhich  the  coefficients  A,  B,  (7,  are  all  finite,  then,  if  h  he- 
comes  infinitesimal,  each  term  after  the  first  is  an  infinitesi- 
mal of  higher  order  than  the  term  p}receding. 

Proof.  The  ratio  of  two  consecutive  terms,  the  third  and 
fourth  for  example,  is 


Dh'  _D 

Ch'  ~  0 


D  . 


By  hypothesis,  Cand  D  are  both  finite;  hence  — -  is  finite; 

hence  when  h  approaches  the  limit  zero,  -^h  becomes  an  in- 

finitesimal  (§13,  Th.  I.).     Thus,  by  definition,  the  term  Dh^ 
is  an  infinitesimal  of  higher  or'ler  than  Ch''. 

Def.  The  orders  of  infinitesimals  are  numbered  by  taking 
some  one  infinitesimal  as  a  base  and  calling  it  an  infinitesi- 
mal of  the  first  order.     Then,  an  infinitesimal  whose  ratio  to 


h 


■''i 
'^ 


!' 


m 


I' 


I   ! 


^    i 


:  P 


!3- 


22 


riC&  DIFFERENTIAL  CALCULUS. 


the  wth  power  of  the  base  approaches  a  finite  limit  is  caUod 
an  infinitesimal  of  the  nth  order. 
Example.     If  h  be  taken  as  the  base,  the  term 

Bh  is  of  cho  first  order  • .  *  Bh  :  h     —  the  finite  quantity  B\ 
Gie    "    "      second"      ','Gli^\W   =        "  "         C; 

EU''  "    "      ?ith       "      • .  •  Eh'' :  7i"  =        "  "         E. 

Cor.  1.  Since  when  ?i  =  0  we  have  Bh""  =  Bh°  =  B  for 
all  values  of  //,  it  follows  that  an  infinitesimal  of  the  order 
zero  is  the  same  as  a  finite  quantity. 

Cor.  2.  It  may  be  shown  in  the  same  way  that  the  product 
of  any  two  infinitesimals  of  the  first  order  is  an  infinitesimal 
of  the  second  order. 

15,  Orders  of  Infinites.  If  the  ratio  of  two  infinite 
quantities  approaches  a  finite  limit,  they  are  called  infinites 
of  the  same  order. 

If  the  ratio  increases  without  limit,  the  greater  term  of  the 
ratio  is  called  an  infinite  of  higher  order  than  the  other. 

Theorem  VII.  In  a  series  of  terms  arranged  according 
to  the  powers  of  x, 

A  +  Bx -\- Cx*  -\-  Dx^  +  etc., 

if  A,  B,  C,  etc.,  are  all  finite,  then,  when  x  becomes  infinite, 
each  term,  after  the  first  is  an  infinite  of  higher  order  than  the 
term  jweceding. 

For,  the  ratio  of  two  consecutive  terms  is  of  the  form  ~^x, 

which  becomes  infinite  with  x  (Th.  III.). 

Def.  Orders  of  infinity  are  numbered  by  taking  some  one 
infinite  as  a  base,  and  calling  it  an  infinite  of  the  first  order. 

Then,  an  infinite  whose  ratio  to  the  ni\\  power  of  the  base 
approaches  a  finite  limit  is  called  an  infinite  of  the  nth.  order. 

Thus,  taking  x  as  the  standard,  when  it  becomes  infinite 
we  call  Bx  infinite  of  the  first  order,  Cx'  of  the  second  order, 
etc. 


LIMITS  AND  INFINITE8IMAL8. 


23 


NOTE  ON  THE  PRECEDING  CHAPTERS. 

In  beginning  the  Calculus,  conceptions  arc  presented  to  the  student 
which  seem  beyond  his  grasp,  and  methods  which  seem  to  lack  rigor. 
Really,  however,  the  fundamental  principle  of  these  methods  is  as  old 
as  Euclid,  and  is  met  with  in  all  works  on  elementary  geometry  which 
treat  of  the  area  of  the  circle.  The  simplest  fonn  in  which  the  princi- 
ple appears  is  seen  in  the  following  case. 

Let  us  have  to  compare  two  quantities  A  and  B,  in  order  to  determine 
whether  they  are  equal.  If  they  are  not  equal,  then  they  must  differ  by 
some  quantity.  If,  now,  taking  any  arbitrary  quantity  h,  we  can  prove 
that 

A-B<n 

without  '.naking  any  supposition  respecting  the  value  of  h,  this  will  show 
that  A  and  B  are  rigorously  equal.  For  if  they  differed  by  the  quantity 
(r,  then  when  7i  was  less  than  a  the  above  inequality  would  not  hold 
true.  But  as  we  have  been  supposed  to  prove  it  for  all  v;ilues  of  h,  it 
must  be  true  when  h  is  less  than  a.  In  this  case  7i  might  be  considered 
an  infinitesimal,  although  in  the  Elements  of  Euclid  it  is  represented  on 
the  page  of  the  book  by  a  figure  nearly  an  inch  square. 

Infinitesimal  quantities  were  formerly  called  infinitely  small.  When 
they  were  introduced  by  Leibnitz  many  able  mathematicians  were  unable 
to  accept  them.  Bishop  Berkeley  wrote  several  essays  against  them,  in 
one  of  which  he  suggested  that  they  might  be  called  the  ghosts  of  departed 
quantities.  The  following  propositions  are  presented  in  the  hope  that 
they  may  save  the  student  unnecessary  efforts  of  thought  in  the  study  of 
this  subject. 

Firstly,  there  is  no  need  that  a  quantity  should  be  considered  as  ab- 
solutely infinite.  A  mathematical  magnitude,  considered  as  .1  quantity, 
must  in  its  very  nature  have  boundaries,  because  mathematics  is  con- 
cerned with  the  relation  between  magnitudes  as  greater  or  less,  and 
we  can  compare  two  magnitudes  as  greater  or  less  only  by  comparing 
their  boundaries.  •  An  absolutely  infinite  magnitude,  having  no  boun- 
daries to  compare,  cannot  be  compared  with  anything. 

Secondly,  it  is  equally  unnecessary  to  suppose  the  existence,  cither  in 
nature  or  in  thought,  of  quantities  which  are  absolutely  smaller  than 
any  finite  quantity  whatever. 


i!l 


:1 


\l 


\  II 


! 


'i      ! 


I 
I      I 


Ml     ^ 


M 


T527  DIFFERENTIAL  CALCULUS. 


But  however  small  a  quantity  may  be,  there  may  always  be  another 
still  smaller  in  any  ratio.  Hence,  although  it  is  perfectly  true  that  no 
quantity  can  be  otherwise  than  finite,  yet  it  is  equally  true  that  a  quantity 
may  be  less  or  greater  than  any  fixed  quantity  we  may  name. 

Both  infinite  and  infinitesimal  quantities  are  therefore  essentially  in- 
definite,  because  by  considering  them  in  the  act  of  increasing  beyond,  or 
decreasing  below,  every  assignable  value,  we  do  away  with  the  very  pos- 
sibility  of  assigning  values  to  them.  They  are  used  only  as  auxiliaries 
to  lead  us  to  a  knowledge  of  finite  quantities,  and  their  magnitudes  arc 
never  themselves  the  object  of  investigation. 

The  essentially  indefinite  nature  of  infinites  and  infinitesimals  may  be 
illustrated  as  follows: 

If  we  have  an  equation  of  the  form 


X  = 


b' 


then  for  every  pair  of  finite  values  we  assign  to  a  and  b  there  will  be  a 
definite  value  of  a;. 

But  if  we  suppose  A  and  5  to  be  infinite,  and  at  the  same  time  inde- 
pendent of  each  other,  there  will  be  no  definite  value  to  x.  Considering 
both  terras  as  absolutely  infinite,  they  will  have  no  bounds,  and  there- 
fore cannot  be  compared  in  value.  Considered  as  increasing  without 
limit,  one  may  be  any  number  of  times  greater  than  the  other,  and  thus 
the  fraction  may  have  any  value  we  choose  to  assign  it.  Seeking  for 
the  value  of  such  a  fraction  is  like  trying  to  answer  the  old  question 
concerning  the  effect  of  an  iiTcsistible  force  acting  upon  an  immovable 
obstacle. 


DIFFERENTIALS  AND  DERIVATIVES. 


25 


CHAPTER 


ere  will  be  a 


OF  DIFFERENTIALS  AND  DERIVATIVES. 

16.  Drf.  An  increment  of  a  variable  is  the  ilifferenco 
between  two  values  of  that  variable. 

An  equivalent  definition  is:  An  increment  is  a  quantity 
juldecl  to  one  value  of  a  variable  in  order  to  obtain  another 
value. 

Notation.  An  increment  is  expressed  by  the  symbol  J 
written  before  the  symbol  of  the  variable. 

Example.     If  we  have  the  different  variables 

X,     y,     u, 
and  the  increments       ^x,     Ay,     Au, 
other  values  of  the  variables  will  be 

X  -f  Ax,     y  -j-  Ay,     u  +  Ati. 

Here  A  is  not  a  factor  multiplying  x,  but  a  symbol  meaning 
^^ increment  of/'  or,  in  familiar  language,  '^^  little  piece  of.'* 

J.i  considering  the  respective  increments  of  an  independent 
variable,  and  of  its  function,  the  following  five  quantities 
come  into  play  and  are  each  to  be  clearly  conceived. 

1.  A  value  of  the  independent  variable,  which  we  may  take 
at  pleasure. 

2.  The  correspondinfj  value  of  the  function,  which  will  be 
fixed  by  that  of  the  independent  variable. 

3.  An  increment  of  the  independent  variable,  also  taken  at 
pleasure. 

4.  The  corresponding  increment  of  the  function,  deter- 
mined by  that  of  the  independent  variable. 

5.  The  ratio  of  these  increments. 


:\\ 


11 


ii 


h  -^ 


1^^ 

i 


i 


THE  DIFFERENTIAL  CALCULUS 

To  represent  those  quantities,  lot  the  relation  between  the 
variable  x  and  the  function  //  bo  expressed  by  a  curve.  Let 
OX  be  one  value  of  x,  and  OX'  another.     Let  XI*  and  X'P' 


Fio.  5. 

be  the  corresponding  values  of  //,  leadincf  to  the  points  P  and 
P'  of  the  curve.     Wo  shall  then  have  — 

1.  OX    —  X,  a  value  of  the  independent  variable. 

2.  XP    =  //,  the  corrcspondini;  value  of  the  function. 

3.  XX  =  z/.r,  an  arbitrary  increment  of  .r. 

4.  EP'  =  A>i,  tio  corresponding  increment  of  //. 

5.  Then,  by  Plane  Trigonometry,  the  quotient  --.--  will  be 

the  tangent  of  the  angle  PQX\  that  is,  the  fangent  of  the 
angle  Avhich  the  secant  PP'  makes  with  the  axis  of  abscissas. 
Thus  we  have  geometrical  representations  of  the  five  fun- 
damental quantities  under  consideration. 

17.  Firf^t  Idea  of  Dijferoifiah  and  Derivatives,     Let  us 
take,  for  illustration,  the  function 

y  =  nx^.  (1) 

Giving  to  x  the  increment  Ax,  the  new  value  of  ;/  will  be 

n{x  +  Ax)\ 

Hence  y  -\-  Ay  =  n(x  +  Ax^  ~  n^""  -f  %nxAx  -f  nAx",      (2) 


niFFKHKNTIA  LS  AND  DERI VA  TI VE8. 


97 


Subtracting  (1)  from  (2),  wo  have,  for  tho  iiicrement  of  y, 

Jy  =  ?i{2x  -f  Jx)Jx,  (8) 

Because,  when  ^x  becomes  inliuitosimal, 

lim.  {2x  +  Jx)  =  2x, 

we  luivo,  for  tho  ratio  of  the  increments. 


-^  =  2nx  -\-  nAXf 
and,  when  Ax  becomes  infinitesimal, 

lim.  -~  =  2wa;. 
nx 


(*) 


(») 


Dcf.  Tho  di£ferential  of  a  quantity  is  its  infinitesimal 
increment;  tliat  is,  its  increment  considered  in  the  act  of  ap- 
proaching zero  as  its  limit,  or  of  becoming  smaller  than  any 
quantity  we  can  name. 

Nofaiion,  The  differential  of  a  quantity  is  indicated  by 
<  he  symbol  d  written  before  the  symbol  of  the  quantity. 

For  example,  the  expressions 

(Ix,    du,    d{x  -\-  y), 

mean  any  infimtesimal  increments  of  x,  ii,  {x  -f-  ?/)>  respect- 
ively. 

Thus  the  substitution  of  d  for  A  in  the  notation  of  incre- 
ments indicates  that  the  increment  represented  by  A  is  sup- 
posed to  be  infinitesimal,  and  that  we  are  to  consider  the  limit 
toward  which  some  quantity  arising  from  the  increment  then 
approaches. 

Using  this  notation,  the  equation  (5)  may  be  written 

f-  =  2«x. 
ax 

We  also  exDtess  this  value  of  the  limiting  ratio  in  the  form 

dy  —  2nxdx', 

meaning  thereby  that  the  ratio  of  the  two  members  of  this 
equation  has  unity  as  its  limit.    This  is  evident  from  Eq.  (3). 


■'■  t 


ihl 


tl 


>»J 


r4^ 


28  TUPJ  DIFFERENTIAL  CALCULUS. 

dv 
Def.     If  7/  is  a  function  of  x,  the  ratio  ~-  of  the  differential 

of  y  to  that  of  x  is  called  the  derivati  76  of  the  function,  or 
the  derived  function. 

18.  Illustrations.  As  the  logic  of  infinitesimals  offers 
great  difficulties  to  the  beginner,  some  illustrations  of  the 
subject  may  be  of  value  to  him. 

Consider  the  following  proposition: 

The  error  introduced  by  neglecting  all  the  powers  of  afi  in- 
crement  above  the  first  may  be  made  as  small  as  we  please  by 
diminishing  the  increment. 

Let  us  suppose  ?i  =  2  in  the  equation  (1).     We  then  have 

the  equations 

y  =  2x'; 

Ay  —  4:xAx  +  3  Jrc'; 
Ay 


(a) 


Ax 


=  4.C  -f  2Ax. 


I'li- 

11 


The  ratio  of  the  two  terms  of  the  second  member  is 

2Ax  Ax 

4.x  '      '^''     W 


■^    ^     Let  us  now  neglect  this  quantity  and  write  the  erroneous 
V  "Iv*.     equation 


V 


<^  ":s- 


Ax 


If,  now, 
we 


suppose 


Ax  < 
Ax  < 
Ax  < 


x 
100' 

X 


J_ 

200 
1 


10000' 


X 


the  equation 
>  {b)  will  still  be  <j  ^qq^q 
true  within 


part; 
part; 
part; 


1000000  J  (^  2000000 

etc.,  etc. 

So  long  as  we  assign  any  definite  value  to  Ax,  it  is  clear 
that  there  will  be  some  error  in  neglecting  Ax.  But  there  is 
no  error  in  the  equations 

dy  =  4:xdx    and     j-  =  4a;, 


nroneous 


DIFFERENTIALS  AND  DFrJVATIVES. 


29 


provided  that  we  interpret  them  as  expressing  the  limit  which 

~  approaches  as  Ax  approaches  the  limit  zero,  and  interpret 

all  our  results  accordingly. 

19.  Illustration  h;  Velocities.  Let  us  consider  what  is 
meant  by  the  familiar  idea  of  a  train  which  may  be  contin- 
ually changing  its  speed  passing  a  certain  point  with  a  certain 
speed.  To  fix  the  ideas,  suppose  the  train  has  just  started 
and  is  every  moment  accelerating  its  speed  in  such  manner 
that  the  total  number  of  feet  it  has  advanced  is  equal  to  the 
square  of  the  number  of  seconds  since  it  started.     Put 

6  =  the  distance  travelled  expressed  in  feet; 
t  =  the  time  expressed  in  seconds. 

We  shall  then  have  d  =  t\ 

and  for  the  distances  travelled: 
Number  of  seconds,  0;    1;    2;    3;      4 
Distance  travelled,     0;     1;    4;     9;     16 
Distance  in  each  second,  1;    3;    5;      7 


111    8    '        5        ' 


B 

I 


11 


1«       2» 


8« 


4& 


6« 


Fio.  6. 


Let  this  line  represent  the  space  travelled  the  first  five 
seconds  from  the  starting  time,  and  let  us  inquire  with  what 
velocity  the  train  passed  the  point  B  at  the  end  of  4*. 

Since  distance  travelled  =  velocity  x  time,  the  mean  ve- 
locity is  found  by  dividing  the  space  by  the  time  required  to 
pass  over  that  space.     Now,  the  train  had  travelled 

16  feet  in  the  time  4  seconds, 
and  (4  -\-  Aty  feet  in  (4  -}-■  At)  seconds, 

or  16  4-  ^At  +  Af  feet  in  (4  -f  At)  seconds. 

Subtracting  16  feel:  and  4  seconds,  we  see  that  in  the  time 
At  after  the  end  of  the  4  seconds  the  train  went  8  J/  -\-  At'* 
E  As  feet.     Hence  its  mean  velocity  from  4*  to  4*  -f-  At  is 


5; 

etc. ; 

25; 

etc.; 

:  i| 

9; 

11;    etc. 

I. 
'^1 

fl 


i 


30  THE  DIFFERENTIAL  CALCULUS. 

--  =  (8  -f-  ^t)  feet  per  second. 

Now  it  is  clear  that,  since  the  train  was  continually  accel- 
erated how  small  soever  we  take  Aty  the  mean  velocity  during 
this  interval  will  exceed  that  witn  which  it  passed  B.  But 
it  is  also  clear  that  by  supposing  At  to  approach  the  limit  zero, 
we  shall  approach  the  required  velocity  as  our  limit.  Hence 
the  velocity  with  which  B  was  passed  is  rigorously 

(Is 


lit 


=  8  feet  per  second. 


Fig.  7. 

20.  Geometrical  IJhistration.  If,  in  the  figure,  we  sup- 
pose the  point  P'  to  approach  P  as  its  limit,  the  increments 
Ax  and  Ay  will  approach  the  limit  zero,  and  the  secant  P' P 
will  approach  the  tangent  at  the  point  P  as  its  limit.  We 
have  already  shown  that 

~  =  tangent  of  angle  made  by  secant  with  axis  of  abscissas. 

Passing  to  the  limit,  we  have  the  rigorous  proposition 

-^  =  tangent  of  angle  which  the  tangent  at  the  point  P 
makes  with  the  axis  of  abscissas. 


DIFFERENTIATION  OF  EXPLICIT  FUNCTIONS.       31 


CHAPTER  IV. 


DIFFERENTIATION  OF  EXPLICIT  FUNCTIONS. 

31.  Def.  The  process  of  finding  the  differentia]  and  the 
derivative  of  a  function  is  called  differentiation. 

As  exemplified  in  §§16,  17,  it  may  be  generalized  as  fol- 
lows:   We  have  given 

(1)  An  independent  variable  E  x. 

(2)  A  function  of  that  variable  =  0(:k). 

(3)  We  assign  to  x  an  increment  e  Ax'j  whereby  (f){x)  is 
changed  into  (f>{x  -f-  Ax). 

(4)  We  thus  have  (f)(x  +  Ax)  —  <p{x)  as  the  increment  of 
(t>{x).    We  may  put 

A(f){x)  =  4>{x  +  Ax)  —  <p{x), 

(5)  We  then  form  the  ratio 

A<p{x) 


Ax 


(a) 


and  seek  its  limit  when  Ax  becomes  infinitesimal.     Using  the 
notation  of  the  last  chaptei',  we  have 


(I(p{x)  A(p{x)  ,  .    .  ^. 


dx  Ax 

which  is  tlic  derivative  of  <p{x). 

In  order  to  find  tlie  ratio  (n),  it  is  necessary  to  develop 
(f)(x  -{-  Ax)  in  powers  of  Ax  to  at  least  the  first  power  of  Ax. 
Let  this  development  bo 

0(.r  +  Ax)  =  X,  -f  X,Ax  +  X,Ax'  -\- .  .  .  .         (1) 

In  the  second  member  of  this  equation  X,,  X,,  etc.,  will  be 
functions  of  x;  and  it  is  evident  that  X^  can  be  nothing  but 


32 


THE  DIFFERENTIAL  CALCULUS. 


h 


0(:?')  itself,  because  it  is  the  value  of  <p{x  -j-  Jx)  when  Jx  =  0. 
Thus  we  have 

J(fr{x)  =  ip{x  +  Ax)  -  (p{x)  =  (X,  +  X,Ax)  Jx  ■[-...  \ 

Passing  to  the  limit, 
d<p{x)  =  Ji^dx; 

M?)  =  X,  (8) 

ax  ^  ^  ' 

Thus,  by  comparing  with  (1),  wo  have  the  following: 

Theorem  I.  7' he  derivative  of  a  function  is  the  coefficient 
of  the  fird  power  of  the  increment  of  the  indejiendent  variable 
when  the  function  is  develojyed  in  potvers  of  that  increment. 

If  we  have  to  differentiate  a  function  of  several  variable 
quantities,  x,  //,  r,  etc.,  we  assign  an  increment  to  each  vari- 
able, and  develop  the  function  in  powers  and  products  of  the 
increments. 

Subtracting  the  original  function,  the  remainder  will  be  its 
increment. 

The  terms  of  highest  order  in  this  increment,  considered 
as  infinitesimals,  are  then  called  the  differential  of  the 
funcbion. 

The  following  are  the  special  cases  by  combining  which  all 
derivatives  of  rational  functions  may  be  found. 

33.  Differentials  of  Sums.  Let  x,  y,  z,  u,  etc.,  be  any 
variables  or  x  auctions  whatever.     Their  sum  will  bo 

x-\-y-\-z-{-n-{-  etc. 

Assigning  to  each  an  increment,  x  will  become  x  -\-  Ax,  y 
will  become  y  -f  Ay,  etc.     Hence  the  sum  will  becom^ 

X  -\-  Ax  -\-  y  -{-  Ay  -\-  z  -{-  Az  -\-  u  -\-  An  -\-  etc. 

Subtracting  the  original  expression,  we  find  the  increment  of 

the  sum  to  be 

Ax  +  Ay  -\-  Az  -\-  Au  ■\-  etc. 


DIFFERENTTATIOIf  OF  EXPLICIT  FUNCTIONS.       83 


msidered 

of 

the 

rhicli 

all 

,  be 

any 

Hence,  when  the  increments  become  infinitesimal, 
(l{x  -{-  y  -{-  z  -\-  u  -{-  etc.)  =  dx  -\-  dy  +  dz  -4-  du  -\-  etc.,  (3) 

or,  in  words: 

Theorem  II.  The  differential  of  the  sum  of  any  number 
of  variables  is  cqtial  to  the  sum  of  their  differentials. 

In  this  theorem  the  quantities  x,  y,  z,  v,  etc.,  maybe  either 
independent  variables,  or  functions  of  one  or  more  variables, 

2li,  Differential  of  a  Multiple.  Let  it  be  required  to  find 
tlie  differential  of 

ax, 
a  being  a  constant. 

Giving  X  the  increment  Ax,  the  expression  will  become 

a{x  +  Ax). 
Then,  proceeding  as  before,  we  find 

d{ax)  =  adx.  (4) 

34,  Theorem  III.  The  differential  of  any  constnut  is 
zero. 

For,  by  definition,  a  constant  is  a  quantity  which  we  sup- 
pose invariable,  and  to  which  wo  cannot,  therefore,  assign  any 
increment. 

We  therefore  have,  from  Theorem  I.  when  x  is  a  variable 
and  ^  is  a  constant, 

d{x  -}-«)  =  dx  +  0  =  dx, 

or,  in  words: 

Theorem  IV.  The  differential  of  the  sum,  of  a  constant 
and  a  variable  is  equal  to  the  differential  of  the  variable  alone. 

Kemark.  It  will  be  readily  seen  that  the  conclusions  of 
§§  22-34  are  equally  true  whether  wo  suppose  the  increments 
to  be  finite  or  infinitesimal.  This  is  no  longer  the  case  when 
powers  or  products  of  some  finit-^  increments  enter  into  the 
expression  for  other  finite  increments. 


il 


I" 


i 


34  THE  DIFFERENTIAL  CALCULUS. 


EXERCISES. 

It  is  required,  by  combining  the  preceding  processes,  to 
form  the  differentials  of  the  following  expressions,  supposing 
a,  h  and  c  to  be  constants,  and  all  the  other  literal  symbols 
to  be  variables. 


I. 

U  —  V. 

2.  2u  —  V. 

3. 

V  -\-x-\-c. 

4.  ax  -)-  by. 

5- 

a^x  -j-  Vy  -f-  c. 

6.  3a;  +  4:ay  -f  3. 

7. 

4:ax  -f-  SJ.-c  —  y. 

8.  Qbx  —  abc. 

9- 

3.r  —  rt  +  «^' 

10.  rtJa;  —  abf. 

II. 

c{2x  -\-  a). 

12.  «(ia:  +  «c)« 

13- 

ac{hu  +  cix). 

14.  Jc(2«a:  —  3J?/). 

15- 

x  —  y  —  z. 

16.   —  n.^'  —  by  —  cz. 

17. 

—  a{bx  —  cy). 

18.   —  J(3aa;  -  3cr). 

19. 

x 
a 

X  4-  y  —  z 

20. =7 . 

0 

21. 

{a -\- h  -\-  c)  {s 

+  /  +  3i^  ■ 

-iy)- 

22. 

{a  +  2b'  -t-  3c' 

)(f 

bv 
a 

cw      abc\ 
'^2         3J' 

25.  Differentials  of  Products  and  Powers.      Take  first 

the  product  of  two  variables,  which  wo  shall  call  u  and  v. 

Then 

Product  =  nv. 

Assigning  the  increments  Au  and  Av,  the  product  becomes 

{u  -\-  All)  {v  -}-  Av)  =  vv  -\-  vAu  -\-  uAv  -f-  AuAv. 

Subtracting  the  original  function,  nv,  we  find 

A{uv)  =  vAu  -{-  (u  -\-  An)  A  v. 

Supposing  the  increments  to  become  infinitesimals,  the  co- 
efficient of  Av  in  the  second  member  will  approach  n  as  its 
limit.     Hence,  passing  to  the  limit  (§  14), 

d{uv)  —  vdn  -j-  ndv. 


DIFFERENTIATION  OF  EXPLICIT  FUNCTIONS.       35 


To  extend  the  result  to  any  number  of  factors,  let  P  be  the 
product  of  all  the  factors  but  one,  and  let  the  remaining  fac- 
tor be  .r,  so  that  we  have 

Product  =:  Px. 
By  what  precedes,  we  have 

a{Px)  =  adP  +  Pdx. 

Supposing  P  to  be  a  product  of  the  two  variables  u  and  v, 
this  result  gives 

d{nvx)  —  xd{vn)  +  uvdx  =  vxdit  -\-  iixdv  -\-  uvdx.      («) 
If  we  add  a  fourth  factor,  y,  we  shall  have 
d{uvxi/)  =  yd{uvx)  -\-  nvxdy. 

If  wo  substitute  for  d{nvx)  its  value  {a),  vv^e  see  that  we 
pass  from  the  one  case  to  the  other  by  (1)  multiplying  all  the 
terms  of  the  first  case  by  the  common  factor  y,  (2)  adding 
the  product  of  dy  into  all  the  other  factors. 

We  are  thus  led  to  the  conclusion: 

Theorem  V.  The  differential  of  the  product  of  any  num- 
hcr  of  variahles  is  equal  to  the  sum  of  the  products  formed  hy 
replacing  each  variable  hy  its  differential. 

Corollary.  If  the  n  factors  are  all  equal,  their  product  will 
become  the  ni\\  power  of  the  variable,  and  the  n  differentials 
will  all  become  equal.  Hence,  when  n  is  an  integer,  we  have 
the  general  formula 

d{x^)  =  x^~^dx  4"  x'^~'^dx  -|-  etc.,  to  n  terms, 
or  J(a;")  =  nx^'^^dx. 

By  combining  the  j)i'eceding  processes  we  may  form  the 
differential  of  any  entire  function  of  any  number  of  variables. 

Examples. 

I.  d{ax  -\-  bxy  -\-  cxyz) 

=  d{ax)  +  d{hxij)  +  d{cxyz)  (Th.  II.  22) 

=  adx  +  bd{xij)  +  cd{xyz)  (Th.  III.  23) 

=  adx  -f-  b{ijdx  +  ^dy)  -\-  c{yzdx  -f-  xzdy  -\-  xydz) 
=  {a  -}-  by  -\-  cyz)dx  -f-  {bx  +  cxz)dy  -f-  cxydz. 


LU.I^U  Ll|<l"    iJlM-J 


I 


\ 


1 


36 


Tim  DIFFERENTIAL  CALCULUS. 


2.  d{ax^  -\-  h)  —  d{ax'') 

=  ad{j-') 

3.  d{ax'y'')  =  ad{x^if) 

=  alifd^x:")  +  :rWO/")] 
=  3(ii/''x''dx  4-  nax^ij  "  ~  ^di/. 


(Th.  IV.) 

(§23) 
(Th.  v.,  Cor.) 

(§33) 

(Th.  V.) 

(Th.  v..  Cor.) 


4.  d{a  4-  x'y  =  7i{a  +  x') "  "  hl{a  +  x')  =  2u{a  +  x') "  "  ^a;</:r. 


EXERCISES. 


Form  the  differentials  of  the  following  expressions,  suppos- 
ing the  letters  of  the  alphabet  from  a  to  n  to  represent  con- 
stants: 

I.  a-\-bx^  •}-  ex*.     A71S,  {2bx -\- 4:cx^)dx. 


2.  B+Cy  +  Df. 

4.  hxyz. 

6.  «(a;'  +  ^uv). 

8.  h{x'y  +  xf). 

10.  bx^y^. 

12.  z{7nx -\- 7iy). 

14.  «(a  —  a;"). 

16.  (a -{- x)  {b  —  y), 

18.  («  —  cc)  {a  —  .r'). 

20.  (^+^:c+(7rc')(2/  +  ;2). 

21.  {A-{-Bf+Cy'){ay+bx) 

23.  («  +  ^?*'')  {ox"^  —  fty^)' 

25.  (a  —  x){b  —  x*)(g  —  x^). 

(?«  +  V). 


3.  aary. 
5.  a{x-\-yz). 
7.  «a;y  -|-  &WV. 
9.  aa^'y'. 
II.  «Z>a;''f/**  +  kn^v\ 

13.  (^•  +  5')  (^  +  5'). 
15.  aa;'  — -  J_?/2;. 

17.  {a -i- x')  {b  -  f). 
19.  .r(a -f  •'*^)  (^  ~  ^')' 

a  ' 
X  —  uv 


22. 


24. 


a 


26. 


rt 


29.   («7/'  -  ?A'6')  (.-c  -  y). 


30 


(I 


IX^         7/ 

bj\a  "^  b 


33.   (rt  +  rry)'. 


27.  x\x''-^y{(i-x)\. 

31.  (rt+a;)'. 

32.  n{a  -\-xy. 
34.  (rta;  +  Z»y)'. 


Th.  IV.) 

(§  ^^3) 

v.,  Cor.) 

(§33) 
(Th.  V.) 
v..  Cor.) 


J,  suppos- 
sent  con- 


DIFFERENTIATION  OF  KXPLIUIT  FUNCTIONS.       'M 

36  Differential  of  a  Quotient  of  Two  Variable.'^.     Let  the 
variables  bo  x  and  y,  and  let  q  be  their  quotient.     Then 

X 

and  qy  =  x. 

Differentiating,  we  have 

ydq  -\-  qdy  =  dx. 
Solving  so  as  to  find  the  value  of  dq. 


Hence: 


,        dx  —  qdy       ydx  —  xdq 
dq  = i-^-  =  ^ '-, 

y  y 


Theorem  VI.  The  differential  of  a  fraction  is  eqtial  to 
the  denominator  into  the  differential  of  the  nuvierator,  minus 
the  numerator  into  the  differential  of  the  denominator ,  divided 
by  tlte  square  of  the  denominator. 

Remark.  If  the  numerator  is  a  constant,  its  differential 
vanishes,  and  we  have  the  general  formula 

d—  = .dx. 

X  X 


m 


EXERCISES. 


)V 


Form  the  differentials  of  the  following  expressions: 

a  -^x 


X 


I. 


a 

+  y 

a 

—  X 

a 

-y 

a 

X 

!• 

a 

-\-bx 

a 

-{-by 

X 

+  y 

x-y 


2. 


(I  +  //' 


X' 

y 


6. 


8. 


lO. 


a 


{b  +  yf 

m  -\-  nx^ 
m  —  nx*' 
mx!^  -|-  ny* 
mz'  —  ny** 


'I 


I 


I 


38 


II. 


U- 


TUE  DlFFEliENTlAL  VALCUim. 


a  .  X  -\-  yz 


a  4-  bx  -\-  cx^* 
m  +  'xy 
VI  —  x^y^' 


a    ,   b 

^     y 


17. 


a 


>^,»* 


xy  4-  x'y 


Tn    ^'  +  -V' 
19.  -5 j. 

x'  -  f 


.    ^m 

V  -j-  a;z* 

1         1 

14. 

5' 

X            X 

16. 

m       n 

?  "  f' 

1        1 

18. 

X       y 

• 

a 

-yn 

x^  -  f 

x'  +  y 


a* 


37.  Differentials  of  Irrational  Expressions.     Let  it  be  re- 
quired to  find  the  differential  of  the  function 


m 


m  and  n  being  positive  integers.     Raising  both  members  of 
the  equation  to  the  nih.  power,  we  have 


?<"  =  x^. 


in   X 
n 


m  — 1 


Taking  the  differentials  of  both  members, 

nu*^~hlti  =  mx"*~\lx, 
whence 

du  _  7n  x^~^  _  m     x^~^ 
7ix~n  u"-^  ~  n  7"^^" -~* 

a  formula  which  corresponds  to  the  corollary  of  Theorem  V., 
where  the  exponent  is  entire. 
Next,  let  the  fractional  exponent  be  negative.     Then 


mn  —  m 
X         ^* 


m     ?*-  -1 

=   -X  » 

n 


,  {<') 


'                                                     _1i       1 

m  ' 

Xn 

and,  by  Th.  VI., 

/  ■5\                 --1 

;                                    ^              dyx'')            m.T"      dx 

\                                      ^^'*  —              VH:    ~        n          2m       ~ 

X      n       dx, 

n 

and,  for  the  derivative. 

dtc           m   -HL-i 
dx          n 

DIFFERENTIATIUxV  OF  EXPLICIT  Fl\NUTWjSti.       39 


,  it  be  re- 


jmbers  of 


From  this  equation  and  from  {a)  wo  conclude: 
Theorem  VII.     The  formula 

d{x^)  =  nx^~^dx 

holds  true  whether  the  exponent  n  is  entire  or  fractional,  yosi- 
(ire  or  negative. 

Wo  thus  derive  the  following  rule  for  forming  the  differen- 
tials of  irrational  expressions: 

Express  the  indicated  roots  by  fractional  exponents,  positive 
or  negative,  and  then  form  the  differential  hy  the  preceding 
methods. 

Examples. 

dx 


I.  d  Va-\-x=^ d{a -\- re)*  =  ^{a -f  x) - Hx  =  - 


2{a-{-xy 


2. 


d^y^  =  d  [*(«  +  ^)  -  *]  =  hd{a  +  x)  -  * 


\hia  -\-x)-  idx  —  —  - — r-T^dx. 

bx 


3.  d{a  +  bx*)^  =  i{a  +  bx')  -  4  Uxdx  = 


I    ,i! 


^1 


EXERCISES. 

eorem  V., 

Form  the  differentials  of  the  following 

expressions: 

I .    Va  -\-  X. 

4.    Va  —  x'. 
a 

2.    Vb  —  x. 

3.    Va  —  bx. 

:hen 

5.    Va  —  bx"". 
b 

Q . 

6.    Vx  -j-  y. 
b 

''    Vx  +  y' 

°'    Va  +  bx*  • 

9-    )/a-bx'' 

10.  {a-\-  x)\. 

II.  {x  —  a)i. 

12.  {}>x*-a)\. 

13.  xVa-{-x. 

14.  X  Va  —  X. 

15.  fVa-by" 

'^dx, 


(l7t 

Find  the  values  of  -7-  in  the  following  cases: 


m 


j6.  u  =  mx  -\ — . 


17.  tc  =  (mx'  —  w)*. 


I       :] 


40 


77//i'  DIFFEIIENTIAL   CALCULUS. 


1 8.  u  =  V((x  +  bx\ 


19.  w 


a 


20.  u  =  X  ^/a  —  X, 

n  4-  X 

22.  u  =  — ■ — , 

a  —  X 


b  -h  ex'' 
21.  u  =  X  i/x'  -{-  a, 
a  —  X 
a  -f-  a; 


38.  Logarilhmic  Functions.    It  is  required  to  difTerentiate 
the  function 

?t  =  log  X, 
Wo  h;ivo 

An  =  log  (r  +  Z/.C)  --  log  x  =  log  '^^-  =  log  (l  -f  -^)- 

It  is  shown  in  Algebra  that  we  have 

log  (1  -I-  h)  =  M{h  -  W  +  W  -  etc.), 
M  being  the  modulus  of  the  system  of  logarithms  employed. 

Hence,  puting  — ^  for  h,  we  find 


/^       iTA/,       1  J.r   ,   1  Ja:'        ,   \ 


and,  passing  to  the  limit. 


du  = ; 

X 


du  _  M 
dx       x' 


In  the  Napcrian  system  M  =  1.  In  algebraic  analysis, 
logarithms  are  always  understood  to  be  Naperian  logarithms 
unless  some  other  system  is  indicated.     Hence  we  write 

J-loff  X      1        ,  ,  dx 

-^~=-;    d-\of!:x  =  — . 

dx  X'  °  X 

Example. 

,  ,  d(axy)       axdy  +  aydx      dy  ,   dx 

fZ-log  axy  =  --^^ — —  = ^-^ — ■■ —  =  --  H . 

^      ^  axy  axy  y        x 

Remark.     We  may  often  change  the  form  of  logarithmic 


3* 

5. 
7. 

9. 

II. 

17. 


Diffe 


DlFFKlttJATlATW.N  OF  KXPUCIT  FUNCTIONS.       41 


rentiate 


fiiuctiouH,  80  as  to  obttiiii  iliuir  (lilTonJutials  in  vuriouH  ways. 
Tlius,  in  tho  last  exaniplo,  wo  havo 

log  {axy)  =  log  a  +  log  x  -\-  log  y, 

from  which  wo  obtain  tho  samo  ditToroiitial  found  abovo.  Tho 
studont  should  find  tho  following  (.lilfurontials  in  two  ways 
when  practicable. 


EXERCISES. 


^^)- 


ployed. 


analysis, 
Tarithms 
rite 


dx 

X 

garithmic 


Differentiate: 

I.  log  {a  -f-  x),  Ans. 

3.  log  {x'  +  b'). 

5.  log  mx. 

7.  log  {ax''  -f-  b), 

9.  log  {x  +  y), 

1 1 .  log  xy. 

13.  \og{a-^b)\ 

^      X  -\-a 
15.   10-.T  — —7. 

17.  ylogx. 


dx 


a-{-  X 


2.  log  {x  —  a). 

4.  log  (.0'  -  b). 

6.  log  7nx^. 

8.  log  m''. 

10.  log  {x  -  y). 

12.  log  (a;' +  ?/'). 

X 

14.  log  -. 

y 

.    ,      a  —  X 

16.  log  T . 

b-y 

18.  log  (r«  —  .r)*". 


29.  Exponential  Functions,     It  is  required  to  differentiate 
the  function 


u  =  «'", 


a  being  a  constant. 
Taking  the  logarithms  of  both  members, 

log  u  =■  X  log  a. 
Diff.Qrentiating,,  wo  have,  by  the  last  article, 

fl?'log  u  =  —  =  dx  log  «. 


,11 


I 


i! 


i 


i    :   '\ 


42  THJ^  DIFFERENTIAL  CALCULUS. 

Hence  du  =  it  log  a  dx  =  a"  log  a  dx', 

which  is  the  required  derivative. 
If  a  is  the  Naperian  base,  whose  value  is 

e  =  2.71828 

we  have  log  «  =  1.     Ilcnce 

d'e" 


s      •       •      a 


dx 


=  c*. 


Hence  the  derivative  of  c*  possesses  the  remarkable  prop- 
erty of  being  identical  with  the  function  itself. 


i 
i 

EXERCISES. 

1                         Di^erentiate: 

I.  a^.    A  us. 

2«"^ 

log  rt  r/.T. 

2. 

«"*. 

3.  c"  +  "- 

4.    ^«--. 

5- 

^^ma:  +  ny  ^ 

6.  /i"**-". 

7.  7i-"*. 

8. 

d'ay. 

9.  a**". 

TO.  cf^b\ 

II. 

ah^'b-K 

12.  e"'  +  «. 

13.  t;^. 

14. 

e«*  +  bv^ 

30.  T'/t''  Trigonometric  Functions. 

The  Sine.     Putting  h  for  the  increment  of  x,  we  have,  by 
Trigonometry, 

sin  (:r  +  h)  —  sin  .t  =  2  cos  (a;  +  i^O  ^"^  i^** 

Now,  let  h  approacli  zoro  as  its  limit.     Then, 
sin  {x  -\-  h)  —  sin  x    becomes     d  sin  x] 
h  becomes  dx,  because  it  is  the  increment  of  x; 
cos  {x  -\-  ^h)  approaches  the  limit  cos  x; 
sin  ^h  approaches  the  limit  ^h  or  ^dx,  because  whe\i 
an  angle  approaches  zero  as  its  limit,  its  ratio   to  i':s  sine 
approaches  unity  as  its  limit  (Trigonometry). 
Hence,  passing  to  the  limit, 

d'sin  X  =  cos  xdx. 


DIFFEUENTIATIOK  OF  EXPLICIT  FUNCTI0:NS.       43 


The  Cosine,     By  Trigonometry, 

cos  {x-\-h)  —  cos  X  =  —  sill  (x  -f  ^A)  sin  \1u 
Hence,  as  in  the  case  of  the  sine, 
d  cos  X  •=■  —  sin  x  dx. 
Taking  the  derivatives,  we  have 
d  sin  X 


dx 

d'COB  X 

dx 


=  cos  x; 


=  —  sm  .r. 


M     N 


Fig.  8. 


PB  =  A  sin  X, 
KP  —  A  cos  x. 


Geometrical  Illustration.     In 
the  figure,  let  OX  be  the  unit-  o 
radius.  Then,  measuring  lengths 
hi  terms  of  this  radius,  we  shall  have 

NK  =  sin  x;     MB  =  sin  (:r  -|-  h) ; 
02^  =  cos  x;     OM  =  cos  {x  i-  Zt); 

Ab  »,  supposing  a  straight  line  from  A' to  B, 

PK  =  -  KP  =  KB  sin  PBK; 
PB  =  KB  cos  PBK. 

When  B  approaches  K  as  its  limit,  the  angle  PBK  ap- 
proaches XOK,  or  Xy  as  its  limit,  and  the  line  KB  becomes 
dx.  Hence,  approaching  the  limit,  we  find  the  same  equa- 
tions as  before  for  d  sin  x  and  d  cos  x. 

It  is  evident  that  so  long  as  the  sine  is  positive,  cos  x  di- 
minishes a«  X  increases,  whence  d'co^  x  must  have  the  nega- 
tive sign. 

The  Tangent.  Expressing  the  tangent  in  terms  of  the  sine 
and  cosine,  we  have 


tan  X  =■ 


sin  X 
cos  x' 


Differentiating  this  fractional  expression, 

cos  xd's'in  X  —  sin  xd'cos  x      sin'  xdx  +  cos"  xdx 


d  tan  X  = 


cos  X 


cos  X 


—  sec''  xdx, 
which  is  the  rerjuired  differential. 


i; 


m 


'  '1 


';   5 


I' 


44 


THE  DIFFEliENTIAL  CALCULUS, 


We  find,  by  a  similar  process. 


,      ,  -  cos  X  ,     -  dx 

a  cot  X  =  d'—. —  =  —  CSC  xdx  = 


c?'sec  x=:  d 


sm  X 
1 


sm  X 


d'Gos  X      sin  xdx 


cos  X  cos*  X 

=  tan  a;  sec  xdx; 
f/'cosec  a;  =  —  cot  a;  esc  xdx. 


cos  » 


Differentiate: 

I.  cos  {a  -f  ?y). 

4.  sin  1/  cos  0. 

7.  sin  (i!:c. 

10.  sin  {h  4"  wr?/). 


EXERCISES. 

2.  sin  {b  —  y). 

5.  tan  2c  cos  v. 

8.  cos  ay. 

ir.  COS  (7i  -1-  my). 


3.  tan  (c  +  z)' 

6.  sin  «  tan  v. 

9.  tan  7/i2;. 

12.  sin  (7i  —  ?w^). 


13.  cos'  X  '  [^Z'cos"^  2;  =  2  cos  xd'coa  a;  =  —  sin  2xdx]. 


14. 
17. 


sm  X. 
sin  X 


15.  sm'  y. 


16.  sin"  7iz. 


18. 


sm  a; 


19. 


cos  X 


COS  ?/  cos  y  '    sm*  ?/ 

20.  Show  that  J(sin'  ?/  -[-  cos'  y)  =  0,  and  show  why  this 
result  ought  to  come  out  by  §  24. 

21.  Differentiate  the  two  members  of  the  identities 

cos  {ci  -\-  y)  =  cos  a  cos  y  —  sin  a  sin  y, 
sin  (a  -\-  z)  =  cos  a  sin  2;  -f-  sin  a  cos  2;, 

and  show  thn^  the  differentials  of  the  two  members  of  each 
equation  arc  identical. 

22.  Show  that  d'log  sin  x  =  cot  x  dx; 

d'log  cos  X  =  —  tan  x  dx. 

31.  Circular  Functions.  A  circular  function  is  the  in- 
verse of  a  trigonometric  function,  the  independent  variable 
being  the  sine,  cosine,  or  other  trigonometri(3  function,  an(i 
the  function  the  angle.     The  notation  is  as  follows: 

If     y  =:  sin  z,      we  write     z  =  sin  ^~  '*  y     or    arc-sin  y; 
If    u  =  tan  X,     we  write    x  =  tan  ^~  ^^  u    or    arc-tan  u; 
etc.  etc,  etc. 


OF' 


BTFFERENTIATTON  OF  EXPLICIT  FUNCTIONS.       4^ 

Differentiation  of  Circular  Functions.     If  we  have  to  dif 
ferentiate 

z  =  sin  (-  *>  y, 
we  shall  have 


y  =  sin  z;    dy  =  cos  z  dz  =  Vl  ::~^£^  dz; 

dy  dy 


.  • .  dz  ~ 


Vl  —  sin'  z       \/\  ~—  y^ 


(a) 


The  Inverse  Cosine.     If  z  be  the  inverse  cosine  of  y,  we 
find,  m  the  same  way. 


dz  =:  -^ 


dy 


The  Inverse  Tangent.     If  we  have 

z  =  tan  ^- ')  ^; 
then,      y  =  tan  .;     r7f/  ...  sec'  ^  rf^  =  (1  +  tan'  ^),?,; 

.'.dz=:^., 

r/^e  /7^^;me  Cotangent.     We  find,  in  a  similar  way, 

^^cot<-ly= ^_ 

^  1  +  y- 

^Ae  /wve.'se  /Sfecaw^.     If  we  have 

z  =  sec  ^~  ^>  ^; 
then,    y  =  sec  z-,    dy  =  tan  z  sec  zdz  =  y  Vf~^^  dz; 


. '.  dz 


__        dy 


yVf~\ 
The  Inverse  Cosecant.     We  find,  in  a  similar  way. 


d-QBG^-^^y 


y  ^y'  ~  1 


(^') 


(^) 


OO 


W 


ill 


r     I 


■li 


Ji 


46 


THE  DIFFERENTIAL  CALCULUS. 


EXERCISES. 

Differentiate  with  respect  to  a;  or  2: 

2.  cos  ^~  *)  (x  +  a). 


I.  sin<~*^  ax. 

3.  sin<~^>  (tnx  -f-  a). 

5.  tan<-«(0-^). 

7.  tan  <-"(-+-). 
\a       xj 

9.  see(-»)  (^+-j. 

II.  sin  ^~  *>  «a;  cos  ^" ') 


X 

a 


4.  cos^  *'  -. 

X 

6.  tan<-«  (2  +  -). 

8.  tan<-»)(a;'). 

10.  sec<~^)  Iz ]. 

1 2.  sec  ^~  *>  a;'  tan  <~  *>  a;. 


Note.  —The  student  will  Hometimes  find  it  convenient  to  invert  the 
function  before  differentiation,  as  we  have  done  in  deducing  the  diflferen- 
tlal  of  sin  (-  »>  x. 

13,  We  have,  by  comparing  the  above  differentials, 

^Z(sin~ ^  y  +  cos" *  ?/)  =  0; 
^7(tan~  \?/  +  cot"  ^  y)  =  0; 
^(sec~ ^  j/  +  CSC"  *  y)  =  0. 

Show  how  these  results  follow  immediately  from  the  defini- 
tion of  complementary  functions  in  trigonometry,  combined 
with  the  theorem  of  §  24  that  the  differential  of  a  constant 
quantity  is  zero. 

33.  Logarithmic  Differentiation.  In  the  case  of  products 
and  exponential  functions,  it  will  often  be  found  that  the  dif- 
ferential is  most  easily  derived  by  differentiating  the  logarithm 
of  the  function.  The  process  is  then  called  logarithmic  dif- 
fcrentiation. 

Example  1.     Find  -.^    when  v  =  a;*"*. 

ax  -^ 


We  have 


log  y  =  m.x  log  x\ 


). 


)• 


1) 


X, 


o  invert  the 
the  dififeren- 

ils. 


the  defini- 
,  combined 
a  constant 

3f  products 
lat  the  dif- 
3  logarithm 
ithmic  dif- 


DIFFERENTIATION  OF  EXPLICIT  FUNCTIONS.       47 

--^  =  7n  loff  X  dx  4-  mdxi 

dy         ,     ^ 

^;;  =  y/("i  log  ^  +  w)  =  7«.r«»'(l  +  log  x). 

Example  2.    y  = 


sm"*  X 


cos"  X 

We  have         log  y  —  m  log  sin  a:  —  n  log  cos  x\ 
dy  _  m  cos  x      n  sin  a; 


ydx         sin  n; 
fZ?/  _  sin  *"  ~  ^  a^ 


cos  X 


dx       cos " + 


—  {m  cos'  x-\-n  sin'  a:). 


a; 


MISCELLANEOUS     ilXERCTSES    IN    DIFFERENTIATION. 

Find  the  derivatives  of  the  following   functions  with  re- 
spect to  a;: 

I.  y  =  X  log  X. 


-4^''*?-  '■£■  =  1 -h  log  X. 


2.  y  =  log  tan  x. 
3'  y  =  log  cot  X. 

X 


Ans.  —  = 


</a;       sin  2a;' 

.        ^Zy  2 

Ans,  -f-= r-—-, 

dx  sin  'Zx 


4.  ?/  = 


S'  y  = 


i/{a'  -  x'y 
x^ 


Ans. 


a' 


^''  y  =  :^ 


(1  +  xY' 


e* +  (?-'"• 


dx       {li'  -  a'')r 
.    ^    fZ?/  _      nx""-^ 
'''  d^  -  (r+"^pr;-i- 


—  .T\9* 


7-  y  =  log(e»'  +  c-'«).         Ans.  ^ 


dx       (6'*-[- e--"") 
dif 


e"  — <?-* 


TT   .    a: 


8.  y  =  lpgtan^  +  ;^  .     Ans.  ^  = 


2 


dy 


cos  .T 


9.  y  = 


a: 


e'"-  1 


f/a: 


rZy  _  c^(l  —  :r)  —  1 


lo.  2^  = 


_  |/(l+a:)+|/(l-a;) 
|/(l+a:)-  i/(l-a:)' 


«a; 


(t'"  -  1)'^ 


X  4/(1  -  x^)' 


!    l! 


l;-  If 


'1 1 

ill 


Al 


i^ 


I 


>   ■   i 


i 

I 


48 


THlil  DIFFERENTIAL  CALCULUS. 


II.   ?/  = 


y 


1+  Vi'^-x') 


12.  y  =  tan  «*. 


13-  ^  =  2;" 


t4.  y  =  sin  (log  a;). 


yl?i5. 


-4w." 


<Jy  ___  ny 

dx 


dy 
dx 

dy 


2V(1 

-  ■^'] 

sec" 

1 

X' 


<^^a'a='. 


^''•'-  Z7  =  ^"(1  +  log  rr). 


15.  y  =  tan 


16.  y  —  log 


—  1 


X 


Vl  -  x'' 

1  +  x\l      1^ 
-x)        2 


Anff. 


Ans. 


tan~ 


dy  _  cos  (log  .1) 
dx  ~         X 

dy  1 


dx         */]_  _ 


X' 


Ans.  — —  = 


dy 


x^ 


17.  ?/  =  log/i/- 


dx       1  —  ic** 


fl  +  a-'-|- 


a; 


i^i  +  a:'  -  a;* 


.        dif  1 


iS.  y~ 


1  --  tan  X 

sec  a; 


19.  y  =  log  (log  x). 


20.  y  =  sin 


—  1 


i  -  X' 


1  + 


X' 


ins. 


uis. 


uis. 


^^        Vl  +  a:'* 
dy  __ 


dx 

dy^^       1 
f^a;       a;  log  a; 

dy  _    -% 


(cos  a;  -[-  sin  a;). 


21. 


y 


=  log  |/^ 


«^cos  a;  —  ^  sin  a: 
a  cos  a:  -f  b  sin  a;* 


IW5. 


dx       1  -|- 


f/y 


a;' 


—  ah 


dx       a"  cos'  x  —  h 


.a  „:^a 


Sin  a: 


22.  If  2/  =  -,  prove  the  relation  -       ^     -_  -f 


7;/ 


r/a; 


Vl  ■\-y^        1/1  + 


=  0. 


a;' 


23-  y  =6-°"^'. 


Ans,  -•-  =  —  2/7* 


dx 


xy. 


DIFFERENTIATION  OF  EXPLICIT  FUNCTIONS.       49 


r 

I 

sin" 

• 

X 

L 

— 

0. 

24.   y 


25-  y 

26.  y 

27.  y 

28.  y  = 

29.  y  = 


><«o    ^^y  _        ^(^  +  a;)  +  ^(^  +  a:) 

(a*  4-  x^)  tan~  *  — .  Ans,  —  =  2x  tan  ~^  — I-  a. 
^  '  a  ax  a 


= Am- 

=  a;  +  log  cos  ^^  - 

=  X  sin  ~  *  X. 
tan  X  tan  ~  *  a;. 


.        dy  1 

(/«  (1   -  X)  fl   _  a;« 

\  A     ^y         2 

J  dx       1 


4-  tan  a;* 


A        dy 

Ans.  -f- 

dx 


dy  __ 


sin  ~  ^  ic  -j — — 


X 


Ans.  -y-  —  sec"  x  tan  ~^a;  -|- 


dx 


tan  X 


30.  y  =  sin  %a:(8in  x^ 


ins. 


dy 

dx 


=  n  (sin  a;)"~^sin  {n  -f-  l)a;. 


31-  y 


_  (sin  nxY 
~  (cos  mxy 


Ans.  -r- 
dx 


dy  _  *"^^  (sill  ^Ja^)  "*~^cos  {mx  —  nx) 


(cos  ?rta;) 


n  +  l 


32.  y  = 


=  e 


•  a'x' 


cos  ra; 


^ws. 


</y  _ 


_  «-oax» 


</a; 


(2rl^a:  cos  rx  -{■  r  sin  ra;). 


33.  y  =  log. 


rt  +  ^  tan 


X 

2 


a  —  &  tan 


ij 


A     ^y 

'Ans.  -r-  =■ 
dx 


ah 


X 


a  cos"  q  —  J"  sin' 


X 


34.  y  = 


a;' 


-4  ms. 


35.  y  =  sm 


.    -i^J  +  l 
V2" 


dy  _ 

dx 

A     ^y 

Ans.   ~  = 


36.  y  =  tan""  *  («  tan  a;).       Ans. 


a;*(l 

-log 

X) 

a;" 

• 

1 

4/1- 

-%x- 
n 

-x" 

dx 

dy^_ 

dx       cos"  x-\-  n^  sin"  x 


iii^i 


iN 


50 


THE  DIFFERENTIAL  CALCULUS. 


37.  y  =  sec 


—  1 


38.  2/  =  (a;  +  «)  tan-  ^  ^/^  -  ^(^0;)^ 


A71S.  -r^  =  tan-^y: 


39.  y  =  sill"  ^  |/(sin  a:). 

2a; 


dx 


40.  y  =  tan 


—  1 


1-x 


a* 


dx 

(iy        2 

dx       1  -\-  X 


1 1/(1  +  cosec  a:). 


.h-\-a  cos  a; 

41.  y  =  sin-* — —-. . 

^  a  -{■  0  cos  a: 


Ans.  -r-  = ^— 

dx 


a  -\-  b  cos  X 


42.  y  =  cos 

43.  y  =  sec 

44.  y  =  tan 


-1 


a;2«  4- 1* 
1    ♦ 


2x'  -  1 
a; 


rfy  2 

-4?is.  -r-  =  

dx 

dy  _ 


V(l  -  xy 
1 


•'^''^-  </a:  ~2(l+a:'0 


33.  Derivatives  tvith  Respect  to  the  Time. —  Velocities.  If 
we  have  a  quantity  which  varies  with  the  time,  so  as  to  have  a 
definite  value  at  each  moment,  but  to  change  its  value  con- 
tinuously from  one  moment  to  another,  that  quantity  is,  by 
definition,  a  function  of  the  time.  We  now  have  the  defini- 
tion: 

If  we  have  a  quantity  (p,  expressed  as  a  function  of  the 

time  =  t,  the  derivative,  -^,  is  the  velocity  of  increase, 

or  rate  of  variation  of  0  at  any  moment. 

This  is  properly  a  definition  of  the  word  velocity;  but  it 
may  be  assumed  that  the  student  has  already  so  clear  a  con- 
ception of  what  a  velocity  is,  that  he  needs  only  to  study  the 
identity  of  this  conception  with  that  of  a  derivative  relatively 
to  t,  which  he  can  do  by  the  illustration  of  §  19. 

The  student  is  recommended  to  draw  a  diagram  to  rep- 
resent the  problem  whenever  hf^  can  do  so. 


DIPFEBBNTIATION  OF  EXPLICIT  PUNCTIOm.       CI 


')■ 


osec  x). 


IS  X 
I 


x')' 


ities.  If 
to  have  a 
due  con- 
ity  is,  by 
le  defini- 

1  of  the 
creasoi 

;  but  it 

It  a  con- 

;udy  the 

datively 

to  rep- 


BXERCISES. 

1.  It  is  found  that  if  a  body  fall  in  a  vacuum  under  the  in- 
fluence of  a  constant  force  of  gravity,  the  distances  through 
which  it  falls  in  the  first,  second,  third,  fourth,  etc.,  second 
of  time  are  proportional  to  the  numbers  of  the  arithmetical 

progression 

1,  3,  5,  7,  etc., 

or,  putting  a  for  the  fall  during  the  first  second,  the  total  fall 

will  be 

fl  +  3«  +  5«  +  7a  +  etc., 

continued  to  as  many  terms  as  there  are  seconds.  It  is  now 
required  to  find,  by  summing  t  terms  of  this  progression,  how 
far  the  body  will  fall  in  t  seconds,  and  then  to  express  its 
velocity  in  terms  of  t,  and  thus  show  that  the  velocity  is 
proportional  to  the  time. 

Ans.  (in  part).  The  total  distance  fallen  in  t  seconds  will  be  aP. 
The  velocity  at  the  end  of  t  seconds  will  be  2at. 

2.  The  above  motion  being  called  uniformly  accelerated, 
prove  this  theorem:  If  a  body  fall  from  a  state  of  rest  with 
a  uniformly  accelerated  velocity  during  any  time  r,  and  if  the 
acceleration  then  ceases,  and  the  body  continue  with  the  uni- 
form velocity  then  acquired,  it  will,  during  the  next  interval 
r,  fall  through  double  the  distance  it  did  during  the  first 
interval. 

Find  (1)  how  far  the  body  falls  in  r  seconds;  (2)  its  velocity  at  the  end 
of  that  time;  (3)  how  far,  with  that  velocity,  it  would  fall  in  another 
interval  of  r  seconds;  then  show  that  (3)  =  2  X  (1). 

3.  The  radius  of  a  circle  increases  uniformly  at  the  rate  of 
m  feet  per  second.  At  what  rate  per  second  will  the  area  be 
increasing  when  the  radius  is  equal  to  r  feet  ? 

Find  (1)  the  expression  for  the  value  of  the  radius  r  at  the  end  of  t 
seconds,  and  (2)  the  area  of  the  circle  at  that  time.    Differentiate  this 

area,  and  then  su  bstitute  for  t  its  value  in  terms  of  r.    Note  that  (<=  —  ). 

\       m/ 

We  shall  thus  have  27rmr  for  the  velocity  of  increase  of  area. 


»' 


11! 


1 


M 


52 


THE  DIFFERENTIAL  CALCULUS. 


4.  A  body  moves  along  the  straight  line  whose  equation  is 

with  a  uniform  velocity  of  n  feet  per  second.     A    ,7hat  rate 
do  its  abscissa  and  ordinate  respectively  increase  'i 

.         2n  -,      n 

Ans.  ~=    and    --=» 

5.  A  man  starts  from  a  point  h  feet  south  of  his  door,  and 
walks  east  at  tlie  rate  of  0  feet  per  second.  At  what  rate  is 
ho  receding  from  his  door  at  the  end  of  f  seconds? 

Ans.  If  we  put  u  =  his  distance  from  his  door,  we  shall 

have 

du  __  c^t 

di~  u' 

6.  A  stone  is  dropped  from  a  point  b  feet  distant  in  a  hori- 
zontal line  from  the  top  of  a  flag-staff  9a  feet  high.  At 
what  rate  is  it  receding  from  the  top  of  the  flag-staff  (1)  after 
it  has  dropped  t  seconds,  and  (2)  when  it  reaches  the  ground, 
assuming  the  same  law  of  falling  as  in  Ex.  1  ? 

At  the  end  of  t  seconds  the  square  of  the  distance  from  the  top  of  the 
flagstaff  —  w'  =  6'  -f  a^t*.    On  reaching  the  ground  we  should  have 


du 
It 


54a« 


7.  The  sides  of  a  rectangle  grow  uniformly,  both  starting 
from  zero,  and  the  one  being  continually  double  the  other. 
Assuming  one  to  grow  at  the  rate  of  m  feet  and  the  other  2m 
feet  per  second,  how  fast  will  the  area  be  growing  at  the  end 
of  1,  2,  10  and  t  seconds?  How  fast,  when  one  side  is  4  and 
the  other  8  feet  ? 

8.  The  sides  of  an  equilateral  triangle  increase  at  the  rate 
of  2  feet  per  second.  At  what  rate  is  the  area  increasing 
when  each  side  is  8  feet  long  ? 

Note  that  the  area  of  the  triangle  whose  sides  =  «  is  -?—  — . 


DIFFKIIENTIATION  OF  EXPLICIT  FUNCTIONS.       ,03 


tion  is 


tiat  rate 


n 

w 

oor,  and 
rate  is 

(76  shall 


Q  a  hori- 

gh.     At 

(1)  after 

ground. 


top  of  the 
mid  have 


I  starting 

other. 

kher  ^m 

the  end 

lis  4  and 

Ithe  rate 
jreasing 


9.  A  man  walks  round  a  lamp,  20  feet  from  it,  koe[)ing 
the  distance  with  x  uniform  motion,  making  one  circuit  per 
minute.  Find  an  expression  for  the  rate  at  which  his  shadow 
travels  on  a  wall  distant  40  feet  from  the  lamp. 

10.  The  hypothcnuse  of  a  right  triangle  is  of  the  constant 
length  of  10  feet,  but  slides  along  the  sides  at  pleasure.  If, 
starting  from  a  moment  when  the  hypothenuso  in  lying  on 
the  base,  the  end  at  the  right  angle  is  gradually  raised  up  at 
the  uniform  rate  of  1  foot  per  second,  find  an  expression  for 
the  rate  at  which  the  other  end  is  sliding  along  the  base  at 
the  end  of  t  seconds,  and  explain  the  imaginary  result  when 
t>10. 

11.  Two  men  start  from  the  same  point,  the  one  going 
north  at  the  rate  of  3  miles  an  hour,  the  other  north-east  5 
miles  an  hour.  Find  the  rate  at  which  they  recede  from  each 
other. 

12.  A  body  slides  down  a  plane  inclined  at  an  angle  of  30° 
to  the  horizon,  at  such  a  rate  that  it  has  slid  3/"  feet  at  the 
end  of  t  seconds.  At  what  rates  is  it  approaching  the  ground 
(1)  at  the  end  of  t  seconds,  and  (2)  after  having  slid  75  feet  ? 

13.  A  line  revolves  around  the  point  {a,  b)  in  the  plane  of 
a  system  of  rectangular  co-ordinate  axes,  making  one  revolu- 
tion per  second.  Express  the  velocity  with  which  its  intersec- 
tion with  each  axis  moves  along  that  axis,  in  terms  of  a,  the 
varying  angle  which  the  line  makes  with  the  axis  of  X 


.       dx 
Ans.  —J 


2b  7t       dy 


dt      sin"  a'     dt 


2a7r 
cos'  a 


14.  A  ship  sailing  east  6  miles  an  hour  sights  another  ship 
7  miles  ahead  sailing  south  8  miles  an  hour.  Find  the  rate 
at  which  the  ships  will  be  approaching  or  receding  from  each 
other  at  the  end  of  20,  30,  60  and  90  minutes,  and  at  the 
end  of  t  hours. 


(d 


'>:t| 


I 


:ii 


64 


TUE  mFFEUEJSTlAL  CALVULUIS 


CHAPTER  V. 

FUNCTIONS   OF   SEVERAL   VARIABLES   AND 
IMPLICIT   FUNCTIONS. 

34.  Def.  A  partial  differential  of  a  function  of  sev- 
eral variables  is  a  differential  formed  by  supposing  one  of  the 
variables  to  change  while  all  the  others  remain  constant. 

The  total  differential  of  a  function  is  its  differential 
when  all  the  variables  which  enter  into  it  are  supposed  to 
change. 

A  partial  derivative  of  a  function  with  respect  to  a 
quantity  is  its  derivative  formed  by  supi)osing  that  quantity 
to  change  while  all  the  others  remain  constant. 

Remark.  The  adjective /?rt?//rtZ  may  be  omitted  when  the 
several  variables  are  entirely  independent. 

Example.    Let  us  have  the  function 

u  =  x'{y  +  2)  +  yz,  (a) 

Differentiating  it  with  respect  to  a;  as  if  y  and  z  wore  con- 
stant, the  result  will  be 

(lit  ■=.  2x{y  -}-  z)dx, 

which  is  the  partial  differential  with  respect  to  x.     Also, 

is  the  partial  derivative  with  respect  to  x. 

In  the  same  way,  supposing  y  alone  to  vary,  we  shall  have 

du  =  {x*  -f-  z)dy,  (c) 

fdu^ 


(b) 


(du\ 
\dyl 


x'  +  2, 


PA  li  TJA  L  DKlil  VA  TI VE8. 


65 


of  sov- 
)  of  the 

Lt. 

3rcntiul 
osod  to 

ct  to  a 
uantity 

len  the 


which  are  tho  partial  difforcntial  aiuJ  durivativo  with  respect 
to  y.  For  the  partial  diilfjreutial  aud  derivative  with  respect 
to  z  wo  have 

du  =  («"  -f  y)dz\  (d) 

Notation  of  Partial  Derivatives.  1.  A  partial  derivative 
is  sometimes  enclosed  in  pa;  jnthoses,  as  wo  have  done  above, 
to  distinguish  it  from  a  total  derivative  (to  be  hereafter  de- 
fined). But  in  most  oases  no  such  distinctive  notation  is 
necessary. 

2.  In  forming  partial  derivatives  the  student  is  recom- 
mended to  use  the  form 

Dm    instead  of    -r-, 
*  dx  * 

because  of  its  simplicity.  It  is  called ///« />j,  o/*?/.  The  equa- 
tions following  {b),  {c)  and  {d)  would  then  be  written: 

D^u  =  ^x{y  +  z)'y 

^^yU     =     X"^    +    Z'y 

D^u  =  x'  -i-y. 


m 


I 


{a) 
Ire  con- 

io. 


EXERCISES. 


have 

(0) 


Find  the  derivatives  of  the  following  functions  with  respect 

to  X,  y  and  z'. 


DyV  =  —x-^2y;    D^v  =  0. 


3.  u  =  x^y'z*. 


1.  V  =  x^  —  xy  +  y^' 

Ans.  D^v  =  2x  —  y; 

2.  w  ^  x'  -\-  x^y  -f-  xz, 
4.  u  =  X  log  y  +  y  log  a;.         5-  ^*  =  (^  +  ^  +  z)\ 

6.  u  —  \/{x  -\-  my).  7.  u  =  {x  +  2?/  -f-  3z)*. 

Note.     In  forms  like  the  last  three,  begin  by  taking  the 
total  differential,  thus: 

du  =  i{x  -\-2y  +  dzy^d  •  {x  +  2y  +  3z) 
=  U^  -\r2y-^3z)-^  {dx  -f-  2dy  +  3dz), 


II 


56 


I'HE  DIFFERENTIAL   CALCULUS. 


Then,  supposing  x  alone  to  vary,  D^u  = 
supposing  y  alone  to  vary,  DyU  = 
supposing  z  alone  to  vary,     D.ti  = 


*• 


2{x+2y+3z) 

1 

{x  +  2y-{-  dzf 
3 

]i{x-{-2y-i-3z) 


i' 


8.  7v  =  {x  -f  y  -\-  ^)^ 

lo.  w  =■  cos  {inx  ~j-  ?/). 

12.  V  —  tan  {x  —  ?/"). 

14.  V  =  cos*  (rt.i'  -j-  Z'^;). 

16.  u  =  xe"  -f-  2/^'"« 


9.  ?y  =  (.t"  +  y'  +  2;')". 

II.  w  =  sin  {x  +  2jy  +  3z). 

13.  ?;  =  sec  (wa;  +  nz). 

15.  y  =  6'*  •■''. 

17.  ?^  =  ic"  -|-  y*. 


iS.    ?i  =  sin  {x-\-y)  GOs{x—y).   19.    u  =  x  sin  y  —  y  sin  ic. 

35.  Fundamental  Theorem.  T'Ac  /o^a^  differential  of 
a  function  of  several  var tables j  all  of  tvhose  derivatives  are 
continuonti,  is  equal  to  the  sum  of  its  partial  differentials. 

As  an  example  of  the  meaning  of  this  theoreni  take  the 
example  of  the  preceding  article;  where  w^e  have  found  three 
separate  differei  als  of  it,  namely,  (b),  (c)  and  (d).  The 
theorem  asserts  .lat  when  x,  y  ani  z  all  three  very,  the  re- 
sulting differential  of  u  will  be  the  sum  of  these  partial  differ- 
entials, namely, 

du  =  2x{y  -f-  z)dx  -j-  (x^  -f-  z)dy  -f  (a;'  +  y)^^» 

To  show  the  truth  of  the  theorem,  let  us  first  consider  any 
function  of  two  variables,  x  and  y, 


u  z=z  (p{x,  y). 


(1) 


Let  us  now  assign  to  x  an  increment  /?.t,  while  //  remains 
unchanged,  and  let  us  call  u'  the  new  value  of  u,  and  ^^.U'  thQ 
resulting  increment  of  u.     We  shall  then  have 


/t^u  =  (p(x  -f  Jx,  y)  -  0(.r,  y). 


(3) 


? 


TOTAL  DIFFERENTIALS. 


57 


32) 


*• 


32) 


i* 


f 


32) 
h32). 

0. 


3m  2;. 

j^mZ  of 
ms  are 
(ds. 

,ke  the 
three 
The 
ne  re- 
differ- 


er  any 


(1) 

emains 
JO  the 


(2) 


In  the  same  way,  if  x  retains  its  vahie  while  y  receives  the 
increment  ^y,  and  if  we  call  z/„w  the  corresponding  incre- 
ment of  w,  we  have 


Jy2i  =  <p{x,  y-^Ay)-  (f){x,  y). 


(3) 


When  Ax  and  Ay  become  infinitesimal,  these  increments 
(2)  and  (3)  become  the  partial  differentials  with  respect  to  x 
and  y. 

Now,  to  get  the  total  increment  of  u,  we  must  suppose  both 
x  and  y  to  loceive  their  increments.  That  is,  instead  of  giv- 
ing y  in  (1)  its  increment  Ay,  we  must  assign  this  increment 
in  (2).  Then  for  the  increment  of  u  we  shall  have,  instead 
of  (3),  the  result 


Ayit'  ■-  (f){x  ^  Axyy  +  Ay)  -  <p{x  +  Ax,  y). 


(4) 


Note  that  (3)  and  (4)  differ  only  in  this:  that  (3)  gives  the 
value  of  Ayii  before  x  has  received  its  increment,  while  (4) 
gives  Ayii,  after  x  has  received  its  increment,  and  is  therefore 
the  rigorous  expression  for  the  increment  of  u  due  to  Ay. 
^  Now,  what  the  theorem  asserts  is  that,  when  the  increments 
become  infinitesimal,  the  ratio  of  Ayu'  to  AyU  approaches 
unity  as  its  limit,  so  that  we  may  use  (3)  instead  of  (4).  To 
show  this,  let  us  put 


^'(^'  ^) = ©• 


Then,  supposing  Ay  to  become  infinitesimal,  and  putting  dyU 
for  that  part  of  the  differential  of  u  arising  from  dy,  we  shall 
have,  from  (3)  and  (4), 


dyU  =  0'(.T,  y)dy; 
dyu'  =  cf)'{x  +  Ax,  y)dy. 


(30 
(4') 


When  Ax  approaches  zero  as  its  limit,  (/)'{x  -f  Ax,  y)  must 
approacli  tlio  limit  0'(.r,  y),  unless  there  is  ji  discontinuity  m 


liifi 


Wl 


68 


TME  DIPFERBm'IAL  CALCULUS, 


the  function  0',  which  case  is  excludod  by  hypothesis.    Thus, 
using  (3')  for  (4'),  we  have 

Total  differential  of  w  s  du  =  f -y-  \dx  -f*  0'(ic,  y)dy 

The  feamd  reasoniilg  ttiay  be  extended  to  the  successive  cases 
of  3,  4,  i  .  .  n  variables. 

The  following  are  examples  of  finding  some  differential  al- 
ready considered  in  Chap.  IV.,  by  this  more  general  process. 


1.    To  differentiate  u  =  xy. 


du 
dx 


du 
'    dy 


y;  --  =  X. 


Total  differential,      du  =  ydx  -}-  ^dy. 


X 


n  \ 


2.     w  =  —  =  xy  ~  ^ 

y 


du 
dx 


du 


y^'y  ay-^y^^y^ 


du  =  y~  ^dx  —  xy~  ^dy  = 

3.       u  =  ax  -\-  bxy  -f-  cxyz, 

du  ,   7      , 

^T;^  =  «  +  ^y  +  cyz; 

du       ,     , 

— -  =  Ox  4-  cxz: 

dy 


_  ydx  —  xdy 


r 


du 

dz 


—  cxy) 


n 


du  r=  (a-\-hy  -{-  cyz)dx  -\-  {hx  -f-  cxz)dy  -f-  cxydz, 
as  in  §  25,  Example  1. 


DIFFERENTIATION  OF  IMPLICIT  FUNCTIONS.       59 


Thus, 


hj 


ve  cases 

itial  al- 
irocess. 


h, 


EXERCISES. 

Write  the  total  differentials  of  the  functiono  given  in  the 
exercises  of  §  34. 

36.  Principles  Involved  in  Partial  Differentiation.  All 
the  processes  of  the  present  chapter  are  aimed  at  the  following 
object:  Any  derivative  expression,  such  as 

du 

dx^    or/>,t., 

presupposes  (1)  that  we  have  the  quantity  u  given,  really  or 
ideally,  as  an  explicit  function  of  .r,  and  perhaps  of  other 
quantities;  (2)  that  we  are  to  get  the  result  of  differentiating 
this  function  according  to  the  rules  of  Chap.  IV.,  supposing 
all  the  quantities  except  x  to  be  constant. 

Now,  because  it  is  often  difficult  or  impossible  to  find  u  as 
an  explicit  functiou  of  x,  we  want  rules  for  finding  the  values 
of  Dg,Uy  which  we  could  get  if  we  had  u  given  as  such  a  func- 
tion of  X.  For  example,  we  might  be  able  to  find  the  equa- 
.tion  u  =  <p{x)  if  we  could  only  solve  one  or  more  algebraic 
equations.  If,  for  any  reason,  we  will  not  or  cannot  solve 
these  equations,  we  may  still  find  D^u  whenever  the  equations 
would  suffice  to  give  ti  as  a  function  of  x  if  we  only  did 
solve  them.  The  following  articles  show  how  this  is  done  in 
all  usual  cases. 

37.  Differentiation  of  Implicit  Functions.  Let  the  rela- 
tion between  y  and  x  be  given  by  an  equation  of  the  form 

(f){x,  ij)  =  0.  (a) 

Representing  this  function  of  x  and  y  by  0,  simply,  and 
supposing  for  the  moment  that  x  and  y  are  independent 
variables,  so  that  0  need  not  be  zero,  we  shall  have,  by  the 
last  section, 

dd)  =  -~dx  +  -j-dy, 
^      dx        '    dy  ^ 


m 


m 


fit  I 


m 


60 


THE  DIFFERENTIAL  CALCULUS. 


But,  introducing  the  condition  that  equation  («)  must  be 
satisfied,  dcf)  must  be  zero,  because  x  and  y  must  so  vary  as  to 
keep  0  constantly  zero.    We  then  find,  from  the  last  equation. 


d(f) 

dy  _ 
dx  ~~ 

dx 
~  d<t) 

dy 

(1) 


which  is  the  required  form  in  the  case  of  an  implicit  function 
of  one  variable. 

Cor.  If  from  an  equation  of  the  form  x  =  f(y)  we  want  to 
derive  the  value  of  D^y,  we  have 

0(a;,  y)=x  -f{y)  =  0; 

_  ^)  ^  _  ^ 

dy  dy  dy' 

dx       dx' 
dy 


^  _         d(^ 

dx~  ~~    ' 


Hence 


Example.     To  find  D^y  from  the  equation 
<Pi^'>  y)  =  y  -ax  =  0. 


We  have 


d<i>  d(p      .      di/ 

—  ::^   fi'       —  ^^  ]_•  '^     —  d' 

dx  *     dy         *     dx         * 


the  same  result  which  we  should  get  by  differentiating  the 
equivalent  equation  y  =  ax. 

Remark.  If  we  should  reduce  the  middle  member  of  (1)  by  clearing 
of  fractions,  the  result  would  be  the  negative  of  the  correct  one.  This 
illustrates  the  fact  that  there  is  no  relation  of  equality  between  the  tA\o 
differentials  of  each  of  the  quantities  x.  y  and  (p,  all  that  we  are  concerned 
with  being  the  limiting  ratios  dy  :  dx\  d(p :  dx,  and  d(p :  dy,  which  limit- 
ing ratios  are  functions  of  x  and  y. 

We  may,  indeed,  if  we  choose,  suppose  the  two  dr's  equal  and  the  two 
(fy's  equal.  But  in  this  case  the  two  rf0's  must  have  opposite  algebraic 
signs,  because  their  sum,  or  the  total  differential  of  0,  is  necessarily  zero. 
Now,  if  we  change  the  sign  of  either  of  the  tZ^'s,  wc  shall  get  a  correct 
result  by  a  fractional  reduction. 


1 


DIFFERENTIATION  OF  IMPLICIT  FUNCTIONS.       61 


ast  be 
J  as  to 
lation. 


(1) 

uction 
irant  to 


ng  the 


clearing 
e.  This 
the  two 
:)ncerned 
oh  limit- 

the  two 
ilgebraic 
'ily  zero, 
a  correct 


EXERCISES. 


Find  the  values  of  -j-,  -j-  or  -j-  from  the  following  equa- 
tions: 

I.  y  —  ax  =  0.  2.  y*  —  yx  -^  x^  =  0. 

3.  x'  +  4-.XZ  4-  z"  =  0.  4.  u{a~x)-^tt'^{b  -\-x)  =  0. 

6.  log  (x-^y)  +  log  (x-y)  =  c. 

8.  sin  ax  —  sin  by  =  c. 

10.  X  {1  —  e  cos  z)  =  a. 


5.  log  a:  +  log  2/  =  c. 
7.  sin  a;  +  sin  y  =  c. 
9.  u  -\-  e  sin  ?*  =  a;. 


38.  Implicit  Functions  of  Several  Variables.  The  pre- 
ceding process  may  be  extended  to  the  case  of  an  implicit 
function  of  any  number  of  variables  in  a  way  which  the 
following  example  will  make  clear. 

Let  u  be  expressed  as  a  function  of  x,  y  and  z  by  the 
equation 

u'  +  xu^  +  {x'  4-  y^)ic  +  a;'  +  y*  +  z'  =  0. 

Since  this  expression  is  constantly  zero,  its  total  differential 
is  zero.     Forming  this  total  differential,  we  have 

(3w'  +  'Hxu  +  a;'  +  y'')du  +  (w'  +  %ux  +  3a;')^a; 

+  {^uy  +  ^^)dy  +  ^z\lz  =  0. 

By  §  34  we  obtain  the  derivative  of  n  with  respect  to  x  by 
supposing  all  the  other  variables  constant;  that  is,  by  putting 
cly  =  0,  dz  =  0,  and  so  with  y  and  z.     Hence 

du        ^  u^  4-  2ux  4-  3x^ 

^^  -  ^x^  -       3^^u  _|_  2ux  +  a;'  +  2/" 

fZ?*       ^  2uf/  -{-  3?/' 

dy  ~     "    ~       3^"  +  %ux  +  a;'  -f-  f' 

du^_  r. 3«^ 

dz  "     '^''  ""       3^"  -h  3wa;  +  a;'  -f  iy'' 


m 


■if 


M 


I 


ipi, 


62 


THE  DIFFERENTIAL  CALCULUS. 


EXERCISES, 

Find  the  derivatives  of  u,  v  or  r  with  respect  to  x,  y  and  z 
from  the  following  equations: 

1.  xu^  +  ifu^  -\~  z*u  =  x^yz» 

2.  a  CC3  {x  —  u)  A^h  sin  {x-\-u)  ^=.  y. 

3.  u'^-\-uy  =  ii\  4.  r*  +  »'  + r*-"  =  r*. 

5.  V  log  X  -\-  z  log  V  =  y.  6.  G^  cos  a;  +  c*  cos  y  =  <;". 

7.  «'  —  3wa;  cos  z  -{-  x'  =  a\     8.  v"  4-  3y.r  cos  2  -f-  a;'  =  Z^'. 

39.  Case  of  Implicit  Functions  expressed  hy  Simvlta- 
ncons  Equations.  If  we  have  two  equations  between  more 
than  two  variables,  such  as 

F^it,  V,  X,  y,  etc.)  =  0,     -^,(w,  v,  x,  y,  etc.)  =  0, 

then,  if  values  01  all  but  two  of  these  variables  are  given,  we 
may,  by  algebraic  methods,  determine  the  values  of  the  two 
which  remain.  We  may  therefore  regard  these  two  as  func- 
tions of  the  others,  the  partial  derivatives  of  which  admit  of 
being  found. 

In  general,  suppose  that  we  have  n  independent  variables, 
a:,,  x^ .  .  .  .r„,  and  m  other  quantities,  n^,  u,  .  .  .  w„,  connected 
with  the  former  by  m  equations  of  the  form 


FXu,,  u,  . 

•   •   '^^m)  -^i)  -^a  •   ' 

.  ^«)  =  0 

F,{u^,  ti,  . 

•   •   '^^m>  "^if  "^"a  • 

.  .  a;„)  =  0 

^m\^\i  ^a  •   •   •   ^m?  *^i*  X^  .   ,   .  X„f  —  {).  J 


(") 


By  solving  these  m  equations  (were  we  able  to  do  so)  we 
should  obtain  the  m  u'b  in  terms  of  the  n  x'8  in  the  form 


u. 


—  0,(^1,  X^  ,  ,  .  X„)', 


•  • 


u, 


m 


—  Y*m(''^j>  X^  .  .  ,  Xf^j'f  ^ 


(*) 


i 


DIFFEBESTxATION  OF  IMPLICIT  FUNCTIONS.      63 


(") 


(*) 


s 


mn  values  of  the  derivatives  -r-*;    -r-*; .  .  . 


and  by  differentiating  these  equations  (b)  we  should  find  the 

— '•    etc 

Now,  the  problem  is  to  find  these  same  derivatives  from  (a) 
without  solving  (a). 

The  method  of  doing  this  is  to  form  the  complete  differen- 
tial of  each  of  the  given  equations  (a),  and  then  to  solve  the 
equations  thus  obtained  with  respect  to  du^,  du,,  etc. 

The  results  of  the  differentiation  may,  by  transposition,  be 

0 

written  in  the  form 

dF,  ,       ,  dF^^       .  ,   dF^  J  dF,  ,  , 

-J—-  du.  4-  -r~  du,  +  .  .  .  +  -7—'  dUn  =  — T"^  dx,    —  etc. : 
du^      '        du^      '  du^  dx^      ^  ' 

dF^  ,       ,  dF^  ^       .  ,   dF,  J  Fd,  ,  , 

-f— ^  du,  4-  -rr--  du.  +  .  .  .  +  -7—  dti^  =  — t^  dx.    —  etc. : 
du^      *        du,      •  dUn  dx^      ' 


dFm.  J         dFm  J      .  , 


dF 


m 


du 


d^K 


dF^ 
dx. 


dx,  —  etc. 


du^       '       du. 

By  solving  these  m  equations  for  the  m  unknown  quantities 
du^,  du, .  .  .  du^j  we  shall  have  results  of  the  form 

du^  =  M^dx^  -\-  M,dx,  +  •  •  •  +  MJlx^', 
du,  =  N^dx^  +  N,dx,  +  .  .  .  +  iV„c?a;„; 
etc.      etc.  etc.  etc.; 

where  M^,  JV„  etc.,  represent  the  functions  of  «,  .  .  ,  u 
x^  .  .  .  Xn,  which  ai'e  formed  in  solving  the  equations. 
We  then  have  for  the  partial  derivatives 

die 


m> 


dx,       ^*^>' 


dx. 


■  =  M,;    etc. 


Example.     From  the  equations 

roos0  =  x,)  (^,j 

r  8m  0  =  y, ) 

it  is  required  to  find  the  derivatives  of  r  and  0  with  respect 
to  X  and  y. 


i 
ill 


fi 


'1! 

-  I 


)  I 


ii 


64 


THE  DIFFERENTIAL  CALCULUS. 


By  differentiation  we  obtain 

cos  ^dr  —  r  sin  Odd  =  dx; 
sin  6dr  -\-  r  cos  6d6  =  dy. 

Multiplying  the  first  equation  by  cos  0  and  the  second  by 
sin  0,  and  adding,  we  eliminate  dO.  Multiplying  the  first  by 
—  sin  0  and  the  second  by  cos  0,  and  adding,  we  eliminate  dr. 
The  resulting  equations  are 

dr  =  cos  0-2i'  -\-  sin  0dy; 
rdd  =  *:'>,3  Air    -  sin  0dx. 

ft. 

Hence,  as  in  the  last  sec.  v>n, 


/^^  _  _  EL?.     (^  _ 
\dxj  ~         r   '     V/v/  ~ 


cos  0 


r 


EXERCISES. 

1,  From  the  equations 

r  sin  6^  =  a;  —  y, 

r  cos  0  =  X  -\-  y, 

find  the  derivatives  of  r  and  0  with  respect  to  x  and  y» 

2,  From  the  equations 

US'"  =  r  cos  0, 
ue~^=  r  sin  ^, 

find  the  derivatives  of  u  and  v  with  respect  to  r  and  0. 

Ans.    £)=i(<3«8ine  +  o-«cos«); 
(^)  =  ^C^"  cos  ^-c- sine); 

0  =  lr<^""^'"''+^"  "»='')• 


■ 


FUNCTIONS  OF  FUNCTIONS. 


00 


0); 


3.  From  the  equations 

w'  -}-  rw  =  a;'  +  y*» 
ru  =  xy, 


u 


find  the  derivatives  of  r  and  u  with  respect  to  x  and  y. 


4.  From  the  equations 
x'-\-y'-\-z'  -  2xyz  -  0, 

n      ;i     dZ  ^  dZ 

find  -7-    and    -7-. 
«a;  ay 


5.  From 

?*'  —  2wz  cos  ^  -f  2;'  =  rt", 
w'  +  221Z  cos  6  -\-  z^  =  Z»', 
-,    ,  r??<      f/?f      fZw     (hv 

^  dz'  dO'  dz'  de' 


40.  Functions  of  Functions.    Let  us  have  an  equa  .or.  of 
the  form 

?*=/(0.  t,  Qy  etc.);  {a) 

where  0,  ?/',  B,  etc.,  are  all  functions  of  x,  admitting  ol  being 
expressed  in  the  form 


0=/iG'^);    ^/'=/,(^);     ^=fz{^)\    etc. 


(*) 


If  any  definite  value  be  assigned  to  .r,  the  values  of  0,  //', 
^'^^  etc.,  will  be  determined  by  (i).  By  substituting  these  val- 
ues in  {a),  u  will  also  be  determined.  Hence  the  equations 
(«)  and  {}})  determine  u  as  a  function  of  x. 

By  substituting  in  (a)  for  0,  ?/',  B,  etc.,  their  algebraic 
expressions /,(:r),  f^{x),  etc.,  we  shall  have  u  as  an  explicit 
function  of  x,  and  can  hence  find  its  derivative  with  respect 
to  X,  But  what  we  want  to  do  is  to  find  an  expression  for 
this  derivative  without  making  this  substitution. 

By  differentiating  («)  we  have 


du  =  -n(l<P  +  TT^^  -f  -Ta^^  +  etc. 
a0        '  df    ^       ^'" 


dO 


By  differentiating  (6), 


d(p  =  -f-dx;    dip  =  -r-dx;    dO  =  -^-d^',    etc. 


ill;  I 


lit 


Tl 


A^ 


66 


TUE  DIFFERENTIAL  CALCULUS. 


^  IT'- 
i  II' 


By  substituting  thoso  values  in  the  last  equation  and  divid- 
ing by  dx,  wu  have 


du  _  du  d(f)      du  dif)       du  dd 
dx  ~  dcj)  dx       dtf)  dx       dO  dx 


(1) 


Tlio  significance  of  this  equation  is  this:  a  change  in  x 
changes  ii  in  as  many  ways  as  there  arc  functions  0,  tp,  0,  etc. 

j-T  -j-fi^  is  the  change  in  ii  through  0; 

-jj  —dx  is  the  change  in  2i  through  tp; 

etc.  etc. 

The  total  differential  is  the  sum  of  all  tliese  separate 
infinitesinic'il  changes,  and  the  derivative  is  the  quotient  of 
this  total  differential  by  dx. 


EXERCISES. 


dti, 


I.  Find  -J--  from  the  equations 

CIX 

u  =  a  sin  {nw  +  w)  +  ^  sin  (mv  —  w); 
V  =  c  -\-  nx;     2u  =  c  —  nx. 

We  find    -r-  =  «w  cos  {mv  4-  w)-\-  hm  cos  imv  —  w)\    -—  =  n; 
(Iv  '         '  'da 


du 
dw 


=  a  cos  {mv  -\-id)  —  b  cos  (mv  —  w); 


dw 
dx^-""' 


Vvhence,  by  the  general  formula, 
du 


dx 


=  an{m  —  1)  cos  {mv  -\-w)-\-  bn{m  -\- 1 )  cos  {mv  —  w). 


2.  Find  -r-  from 
dx 


u  =  c^  -{-  e'f'; 

0  =  e^',    fp  =  ne 


—  X 


Ans.  e^+^-we"^-*. 


•?.  Find  -,    from 
dy 

v"  4-  ^'0  +  V-''  =  «; 


0  =  wi(a  +  7/);     ^\y  -  ny. 


1 


FUNCTIONS  OF  FUNCTIONS, 


67 


(1) 


X 


4.  Find  -}-  from 


dz 


r  COB  X  —  r  ^uv  X  —  a  —  y*y 
X  =  mz  -{-  h;    y  =■  cos  nz. 


5.  Find  V  from 


dz 


r*  +  xr^  +  2/V  +  0'  =  0; 

x*  -{•  az  =  0;    y'  -f  rtz'  =  0;     0  =  wa;. 


41.  The  foregoing  tlieory  applies  equally  to  the  case  in 
which  the  function  is  one  of  two  or  more  variables,  some  of 
which  are  functions  of  the  others.     For  example,  if 

u  =  <p{x,  z),  (n) 

then,  whatever  be  the  relation  between  x  and  z,  we  shall 
always  have,  for  the  complete  differential  of  u, 


^"^ = ©''^ + (S''^- 


Suppose  that  x  is  itself  a  function  of  z.     We  then  have 

dx  =  -7-  dz. 
dz 

By  substitution  in  the  first  equation  we  have 


du  = 


du\dx       fduY 
dx  Idz        \dzl 


dz; 


du  _  /du  \  ^c   I    (du  \ 
'  '  dz  ~  \dxJ  dz       \dz}' 


(*) 


du 


The  two  values  of  -j-  which  enter  into  this  equation  are 

different  quantities.  A  change  in  z  produces  a  change  in  u 
in  two  ways:  first,  directly,  through  the  change  in  z  as  it 
appears  in  {a)]  second,  indirectly ,  by  changing  the  values  of 

X  in  (a).     The  first  change  depends  upon  \-f]  in  the  second 


j| 


TUE  DIFFKltKNTIAL  CALCULUS. 


du\  dx 


C8 


member  of  (/;);  the  second  upouf-^J  -v-;  while  the  first  mem- 
ber of  (/>)  expresses  the  total  change. 

It  is  in  distinguisliing  the  two  vahics  of  a  derivative  thus 
obtained  that  the  terms  pdrfial  derivative  and  total  derivative 
become  necessary.     If  we  have  a  function  of  the  form 

u  =f{x,  y,to...  z), 

in  which  any  or  all  of  the  quantities  x,  ?/,  m,  etc.,  may  be 
functions  of  z,  then  the  partial  derivative  of  u  witli  respect 
to  z  means  the  derivative  when  we  take  no  account  of  the 
variations  of  x,  y,  tv,  etc.;  and  the  total  derivative,  with 
respect  to  z,  is  the  derivative  when  all  these  variations  are 
taken  into  account. 

In  such  cases  the  partial  derivative  has  to  be  distinguished 
])y  being  enclosed  in  parentheses  (§  34).  This  is  why  the  last 
equation  is  written 

du  _  (du\       Uhi\  dx 
dz  ~  \dzl       \dxl  dz' 

42.  Extension  of  the  Principle.  The  principle  involved 
in  the  preceding  discussion  may  be  extended  to  the  case  of 
any  number  of  independent  variables  and  any  number  of 
functions.    If  we  have 

r  =  (p{u,  V,  w  .  .  .  X,  y,  z  .  .  .), 

in  which  x,  y,  z,  etc.,  are  the  independent  variables,  while 
iCf  V,  w,  etc.,  are  functions  of  these  variables,  we  shall  have 

Then,  since  u,  v,  zv,  etc.,  arc  functions  of  x,  y,  z,  etc.,  wo 
have 

du  =  -y-dx  +  -j-dy  +  etc. ; 
dx  dy 

dv  =  -j-dx  4-  -j-dy  +  etc, 
dx       ^   dy  ^ 


fin 


FUNCTIONS  OF  FUNCTIONS. 


CO 


By  subsiitutiug  those  values  in  the  preceding  equation  we 
find* 


<lr  = 


+ 


[©-i-(3:l+(a:^+--> 

_\(ly)  '''  [dul  dy  "^  \ 


^/0\  do   .  ~1  , 


+ 


TTenco,  writing  r  for  0,  its  equivalent, 

r       [dr\    ,    f  dr\dic    ,    (dr\dif   ,     , 


dx 

etc.      etc. 


etc. 


etc. 


EXERCISES. 

The  independent  variables  r  and  ^  being  connected  with  x 
and  y  by  the  equations 

.'c  =  r  cos  ^, 
y  =  r  sin  ^, 

it  is  required  to  find  the  derivatives  of  the  following  functions 
of  X,  y,  r  and  8  with  respect  to  ?'  and  0.  We  call  each  of  the 
functions  ti. 


I.  11  z=  0'^  -\-  2xy  cos  36'. 

Here  we  have 

©  =  -= 

idu\ 
\do) 

f^  =  3Z/  cos  20; 
ax 

dii 
dy 

dx            . 
-—  =  cos  0; 

dy 
dr 

-y, 

dy 

dO 

=  2a!  cos  20; 

=  sin  0 ; 

=  r  cos  0  =  2!. 


I 

I 

'I 


*  Here,  wlien  we  use  the  symbol  cj)  instead  of  r,  there  is  really  no 
need  of  enclosing  the  partial  derivatives  in  parentheses.   We  have  done 


it  only  for  the  convenience  of  the  student. 


ilf 


-  J 


hi 


70 


TJT£?  DIFFERENTIAL  CALCULUS. 


rr  ^^^  _  /^'^\  j^du  dx      du  dy 

dr  ~~  \dri    '   <?«  ar       dv  dr 


dy 

=  2r-\-2y  cos  0  cos  20  -{-  2a;  sin  0  cos  20 
=  2r(l  +  cos  20  sin  20)  =  r(2  +  sin  40); 


and,  in  the  same  way, 

~  ^  2r'  cos  40, 
aO 

We  might  have  got  the  same  result,  and  that  more  simply,  by  sub- 
stituting for  X  and  y  in  the  g'.veu  equation  their  values  in  terms  of  r  and 
0.  But  in  the  case  of  implicit  functions  this  substitution  cannot  be 
made;  it  is  therefore  necessary  to  be  fami^ar  with  the  above  method. 


2  3  3 

a     ,   a;  —  y 

2.  u  =  -,-\ i--^ 

?•  a 

_  a'      b'      2ab 

3*    ^*  —  "~a  ~r  ""a  a~* 

X       y         r 
4.  u  =  r^  —  {x  —  yy. 

1 


cos  2(^. 


5-  'if'  = 
6.  u  = 


a;  siu  2^  +  ?/  cos  2/^* 

1 1__ 

a;  cos  26^      ?/  sin  20' 


7.  ^^  =  7'^  -f-  x^ 


f 


Let  V  and  ?<;  be  given  as  implicit  functions  of  p  and  6  by 
the  equations 


w  =  «!>; 


V' 


I 
-j-  w^  —  2p  sin  6.  \ 


•  •  •  • 


(«) 


It  is  required  to  find  the  total  derivatives  of  the  following 
functions  with  respect  to  p  and  6  respectively: 


8.  u  =^  v"^  -{-  w' 
nh 


10.  n  = 


vw 


(J,  u  =  v^  —  2vw  cos  0  -\-  id^, 
II.  u-=^  {v  -\-  w)  sin  6, 


12.  u  =  {v  —  w)  cos  ^, 

13.  7^  =  'w^  —  v''  -[-  2{w  -f-  ?')p  cos  (^. 


1 


J 


PARTIAL  DEIilVATIVES. 


71 


by  sub- 
of  r  and 
innot  be 
thod. 


f 


id  ^by 

.     («) 

lowing 


From  the  pair  of  equations  (a 

we 

find 

dv 

V 

dw 

w 

dp 

~V 

dp 

~2p' 

dv 
dO 

=  ivc. 

■~0; 

dw   , 
dO''' 

ziiW  cot  0; 

which  values  are  to  be  substituted  in  the  symbolic  partial  derivatives  of  u. 

43.  Remarks  on  the  Nomendature  of  Partial  Derivatives* 
There  is  much  diversity  among  mathematicians  in  the  no- 
menclature perta-ning  to  this  subject.  Thus,  the  term  "  par- 
tial derivative^'  is  sometimes  eii:tended  to  all  cases  of  a  deriva- 
tive of  a  function  of  several  variables,  with  respect  to  any  one 
of  those  variables,  though  there  is  then  nothing  to  distinguish 
it  from  a  total  derivative. 

Again,  Jacobi  "  nd  other  German  writers  put  the  total  deri- 
vatives in  pareni.'.oses  and  omit  the  latter  from  the  partial 
ones,  thus  reversing  the  above  notation. 

If  we  have  to  express  the  derivative  of  0(.r,  y,  z,  etc.)  with 
respect  to  z,  the  English  writers  commonly  use  the  symbol 


-y  in  order  to  avoid  writing  a  cumbrous  fraction. 
have  such  forms  as 


We  thus 


d  [x 
dx 


7)  f^' _L -^  4- ^"V 


each  of  which  means  the  derivative  of  the  expression  in  paren- 
theses with  respect  to  x,  and  which  the  student  can  use  at 
pleasure. 

44.  Dependence  of  the  Derivative  upon  the  Form  of  the 
Function.  Let  x  and  y  be  two  variables  entirely  independent 
of  each  other,  and 

u  —  0(.r,  y)  (a) 

a  function  of  these  variables.  AVithout  making  any  change 
in  u  or  .r,  let  us  introduce,  instead  of  y,  another  independent 


m 


i      ! 


72 

variable 
making 


%,  s 


y  '■'i 


the 


TUE  DIFFERENTIAL  CALCULUS. 

upposed  to  be  a  function  of  x  and  y.    Then,  after 
substitution,  we  shall  have  a  result  of  the  form 


?f 


=  F{x,  z). 


iP) 


Now,  it  is  to  be  noted  that  although  both  u  and  x  have  the 

du 
same  meaning  in  {b)  as  in  {a),  the  value  of  -j-;  will  be  differ- 

ent  in  the  two  cases.     The  reason  is  that  in  {a)  y  is  supposed 
constant  when  we  differentiate  with  respect  to  x,  while  in  {b) 
it  is  z  which  is  supposed  constant. 
Analytic  Illustration.     Let  us  have 

u  =  ax^  -\-  by^. 
du 


This  gives 


dx 


=  2ax. 


(c) 


Let  us  now  substitute  for  y  another  quantity,  z,  determined 
by  the  equation 

z  =  y  -{-  X    or    y  =  z  —  X. 

We  then  have  ^i  =  ax-^  -\'b{z  —  xY', 

.    —-  =  'Zax  -f-  '^b{x  —  7)\ 

which  is  different  from  {c). 

Our  general  conclusion  is:  The  jjnrtial  derivative  of  one 
variable  luith  respect  to  another  depends  not  only  upo}i  the  re- 
lation of  those  two  variables,  but  upon  their  relations  to  the 
variables  which  we  sup- 
pose constant  in  differen- 
tiating. 

Geometrical  Illustra- 
tion. Let  r  and  B  be  the 
polar  co-ordinates  of  a 
point  P,  and  x  and  ?/  its 
rectangular  co-ordinates. 

Then  Fig.  9. 


X  =  r  cos  B\ 
7/  =  r  sin  8) 

r'  =  ic'  +  y\ 


(d) 


fr 
th 
th 
m( 
fn 
Tl: 
no] 
gre 


PARTIAL  DERIVATIVES. 


78 


after 
>rm 

(*) 

ve  the 

diller- 

3pose<l 
in  (b) 


rmiiied 


Regarding  r  as  a  function  of  x  and  y,  we  have 

dr      X  ^ 

—-=  -  =  cos  0. 
ax      r 


w 


But  we  may  equally  express  r  as  a  function  of  a:  and  6,  thus: 

r  =  :c  sec  0,  (/) 

dr 


We  then  have 


dx 


=  sec  0. 


(9) 


Referring  to  the  figure,  it  will  be  seen  that  we  derive  (e) 
from  (d)  by  supposing  x  to  vary  while  y  remains  constant; 
that  is,  by  giving  the  point  P  an  infinitesimal  motion  along 
the  line  FQ  \\  to  OX.  In  this  case  it  is  plain  that  the  incre- 
ment of  r  (SQ)  is  less  than  that  of  x.  But  in  deriving  (//) 
from  (/)  we  suppose  x  to  vary  while  6  remsiins  constant. 
This  carries  the  point  P  along  the  straight  line  OPE;  and 
now  it  is  evident  that  the  resulting  increment  of  r  (PB)  is 
greater  than  that  of  x. 


of  one 
the  ve- 
to the 


n 


Q 


id) 


74 


THE  DIFFERENTIAL  CALCULUS. 


I' 


lii 


CHAPTER  VI. 


DERIVATIVES   OF   HIGHER   ORDERS. 


4:5.  If  we  have  given  a  fuuction  of  x, 


dy 


we  mav,  by  differentiation,  find  a  value  of  —-.     This  value 

'       -^  dx 

will,  in  general,  be  another  function  of  Xy  which  we  may  call 
(})'{x).     Thus  we  shall  have 


dy 
dx 


0'(;r). 


Now,  this  function  0'  may  itself  be  differentiated.     If  we 
call  its  derived  function  0'', 
we  shall  have 


a 


All 

(U   _  d(fy'{x) 


dx 


dx 


0"(.r).  (^0 


<» 


AX 


AX 


Let  us  examine  the  geo- 
metrical meaning  of  this 
equation,  by  plotting  the 
curve  representing  the  origi- 
nal equation  y  =  (p  (x). 

Let  X,  x'  and  x"  be  three 
equidistant  valucM  of  the  ab-' 
scl-ii-'a,  so  that  the  increments 
x/  —  X  and  x"  -  x/  ee  -dx  are 
couo-.  Let  7*,  Q  and  R  bo 
the  corr<^sp!.n;ling  points  of  the  curve.  Let  //,  ?/'  and  y 
be  the  tiiri,u  coiTes])ond:iig  values  of  y. 


Xo     ?ci     yi2 


Fio.  10. 


n 


^ 


f>] 


-.-  i 


lis  value 
may  call 


.     If  we 


II 


;m 


11 


DERIVATIVKS  OF  EIOnER  ORDERS. 


75 


» 


f'J 


Then  we  may  put 


Au^y'  ~y    =  MQ, 
A'y  =  y''-y'  =  NR, 

as  the  two  correspondirg  increments  of  y. 

It  is  evident  that  these  increments  will  not,  in  general, 
be  equal;  in  fact,  that  they  can  be  equal  only  when  the  thre*^ 
points  of  the  curve  are  in  the  same  straight  line.  If  D  is  the 
point  in  which  tlie  line  PQ  meets  the  ordinate  of  Ry  then 
DR  will  be  the  difference  between  the  two  values  of  Ay,  so 
that  we  shall  have 

DR  —  A'y  —  Ay  —  increment  of  Ay. 

Hence,  again  using  the  sign  A  to  mark  an  increment,  we 

shall  have 

DR  =  A  Ay  =  A'y,  (h) 

in  which  the  exponent  does  not  indicate  a  square,  but  merely 
the  repetition  of  the  symbol  A. 

Theorem  I.  When  Ax  becomes  infinitesimal,  A'^y  becomes 
an  infinitesimal  of  the  second  order. 

For,  if  D  be  the  point  in  which  PQ  produced  cuts  the 
ordinate  X^R,  we  shall  have,  in  the  triangle  QRD, 


7-.  71       ^  Tasini?  07)        .. 


(^) 


sin  QRD 

When  Ax  becomes  an  infinitesimal  of  the  first  order,  so  do 
both  QD  and  the  angle  RQD,  but  the  angle  QRD  will  remain 
finite,  because  it  will  approach  the  angle  QDN  as  its  limit. 
Hence  the  expression  will  contain  as  a  factor  the  product  of 
two  infinitesimals  of  the  first  order,  and  so  will  be  an  infini- 
tesimal of  the  second  order. 

Since  both  the  quantities  QD  nnd  RQD  depend  upon  Ax, 
we  conclude  that  the  ratio 

Ay 

Ax' 
may  remf.',Hi  UrAu''  when  Ax  l)ecomes  infinitesimal.     In  fact. 


M 


76 


THE  DIFFERENTIAL  CALCULUS. 


f 


from  the  way  we  have  formed  these  quantities,  we  have 

lim.  2^.  =  lim. -^  =  ^^  =  0"(^). 
Hence — 

Theorem  II.  If  we  take  tiuo  equal  consecutive  infinitesimal 
increme7its,  =  dx,  of  the  independent  variable,  then — 

1.  The  difference  between  the  corresponding  infinitesimal 
increments  of  the  function  divided  by  dx^  will  approach  a 
certain  limit. 

3.  This  limit  is  the  derivative  of  the  derivative  of  the 
function. 

Def.  The  derivative  of  the  derivative  is  called  the  second 
derivative. 

The  derivative  of  the  second  derivative  is  called  the  third 
derivative,  and  so  on  indefinitely. 

Notation.  The  successive  derivatives  of  y  with  respect  to 
X  are  written 

dy      d^u     d^xi 

dx'     dx''     dx''    ^^^'' 


I  ? 


or 


I^xV;      DxVr      J^xVy      etc. 


46.  Derivatives  cf  any  Order.  The  results  we  have 
reached  in  the  last  article  may  be  expressed  thus:  If  we  have 
an  equation 

y  =  <P{^), 

the  first  der'vf^tive  is  given  by  the  equation 


dx 


=  0'W. 


Then,  by  differentiating  this  equation,  "we  have,  by  the  last 
theorem, 


d.^r 


'■x  _  d*y 


dx 


dx" 


0"W. 


m] 


lave 


Initesimal 

Initemnal 
iproach  a 

ve  of  the 

3  second 

;he  third 

respect  to 


we    have 
f  we  have 


the  last 


m 


DERIVATIVES  OF  IIIGUEIt  OIWEIiS. 
Again,  taking  the  derivative,  we  have 

and  wo  may  continue  the  process  indefinitely. 

EXERCISES    AND    EXAMPLES. 

I.  To  find  the  successive  derivatives  of  ax\ 

ax  ' 

d'y 


77 


dx 


3   =  Gn; 


and  all  the  liighcr  derivatives  will  vanish. 

Form  the  derivatives  to  the  third,  fourth  or  nth  order  of— 


4.  {(I  +  xy. 

7-   (a-  x)~ 


2.  ax.  3.  ^,^.-1^ 

5-   i'i-xy.  6.   (a-^x)-\ 

8.  {a'-j-xy.  9.   2a'x'-\-x\ 

10.  a-{-  bx  +  vx'  +  Ju^  +  he". 

11.  1  +  a;  +  x'  +  .,^  +  .,•  +  .^•^  +  .  .  .  _^  .^«, 

^^-   \-  ^^  +  ^^  -^^^  +  ^^^  -  .^^  -I-  .  .  .  -\-  (-  I)",.., 

'^-   ''"'  ^4.   .^(  .5.    (^/  +  .r)^  ,6.   (r. +  :,•)?. 

17.  If        //  =  c^  find  A"//  =  (f{\og  (f)\ 

iS.  From  .y  r=  m.e^  find  the  ;/tli  derivative. 

19.  From  //  =:  vie"'  show  that  D,'\//  =  h'^ij. 

Find  the  first  three  derivatives  of  the  expressions: 

^°*  ^  •  21.  ax^\  22.  x^" 

^^'  ^^S  ''•  24.   log  {a  +  .r).        25.   m  log  x. 

26.  log  (r.  -  X).  27.   log  (r,,  +  mx).     .8.  log  («  -  rnx). 

29.  Show  that  if  ?/  r=  sin  x,  then  ^^  =  -  y 

dx^  •'' 


f 


ri  ''i 


78 


THE  DIFFEBENTIAL  CALCULUS. 


30.  Show  that  tho  same  equations  hold  true  ii  y  =  cos  z 
or  it  y  =  a  cos  x  -{-  b  Bin  x. 

31.  Find  the  law  of  formation  of  the  successive  derivatives 
of  sin  mx  and  cos  mx. 

Especially,  the  {n  -\-  4)th  derivative  =  wth  der.  x  what? 
{n  -f  2)th  derivative  =  wth  der.  x  what? 

32.  Find  the  /ith  derivative  of  c"**. 

33.  Find  three  derivatives  of  c*"*  sin  nz, 

34.  If  u  =  ^/^  show  that  ~  =  (1  +  log  y)'-^  +  ^. 

35.  Find  two  derivatives  of  u  =  tan  z. 

36.  Find  two  derivatives  of  u  =■  cos'  z. 

37.  Find  two  derivatives  oi  10  --  sec"  z. 

T,^.  Find  two  derivatives  of  u  =  cos"  ;2  —  sin'  z. 

39.  Find  two  derivatives  of  u  =  cos  2^;. 

40.  Find  two  derivatives  of  u  =  e~ ""'. 

41.  Find  two  derivatives  of  w  =  sin  ^"  ^>:r. 

4*7.  Special  Forms  of  Derivatives  of  Circular  and  Ex- 
ponential Functions.     Because 

cos  X  =  sin  (x  4-  ^tt)     and     —  sin  x  =  cos  {x  -f-  i^r), 

the  derivativet:  of  sin  x  and  cos  .1*  may  be  written  in  the  form 

Djg  sin  ./;  =  sin  [x  -\-  ^n) 
and  /)x  cos  x  =  cos  (.?;  +  ^tt). 

Hence,   the  sine  and  cosi^ie  are  such  functions  that   their 
derivatives  are  formed  by  increasing  their  argument  by  ^tt. 
Differentiating  by  this  rule  71  times  in  succession,  we  have 

^"  sin  X 


Dj"  sin  x  = 


DJ*  cos  X  — 


fZ"  COS  X 


sin  \x-\-^^7rj; 


n 


cos(:r  +  -;r); 


W. 


results  which  can  be  reduced  to  the  forms  found  in  Exercises 
29  and  30  preceding. 


=  COS  X 

rivatives 

X  what? 
X  what? 


and  Ex- 
he  form 


id    their 
\Q  have 


ixercises 


f 


m 


DERIVATIVES  OF  UIGIIEli  0RDER8. 


79 


48.  Successive  Derivatives  of  an  Implicit  Function.  If 
the  relation  between  y,  the  function,  and  x,  the  independent 
variable,  is  given  in  the  implicit  form 

f(x,  y)  =  0, 

then,  putting  u  for  this  expression,  we  have  found  the  first 
derivative  to  be 

du 
dy  _       dx  .  . 

dx  ~       Sw'  ^  ' 

Tiy 

The  values  of  both  the  numerator  and  denominator  of  the 
second  member  of  this  equation  will  be  func^tions  of  x  and  2/, 
which  we  may  call  X^  and  Y,.     Wo  therefore  write 


dy 
dx 


Y 


(b) 


Differentiating  this  with  respect  to  x,  we  shall  have 
d'y  _ 


'  dx  "^     '  dx 


dx' 


y; 


(c) 


X^  and  Yf  being  functions  of  both  x  and  ?/,  we  have  (§  41) 

dX,  ^  (d,X\       IdXXdji. 
dx        \  dx  I       \  dy  Idx^ 

dY,  ^  (^A    ,    I^^sYM 
dx        \  dx  1       \  dii  Idx' 


dy 


Substituting  in  these  equations  the  values  of  ~  from  (J), 

and  then  substituting  the  results  in  {c),  we  shall  have  the  re- 
quired second  derivative. 

The,  process  may  then  be  repeated  indefinitely,  and  thus 
the  derivatives  of  any  orders  be  found. 

Example.  Find  the  successive  derivatives  of  y  with  re- 
spect to  X  from  the  equation 

X*  —  xy  -j-  ^'  E  ?^  =  0, 


11 


'(| 


V 


: 


i       I 


80  TUB  DIFFERENTIAL  CALCULUS. 

Wehave  ^|  =  2:.  -  y;     ^^  =  -  a:  +  gy; 

(1^1  ^  2a;  -  y, 
rtfa;       a:  —  2y' 

which  is  a  special  case  of  {a)  and  (/>),  and  where 

X^  =  2.6'  —  y/     and     Y^  =  —  x  -\-  2fj. 

Differentiating  tlie  equation  (a'),  wo  have 

d^y       ^''      ^''^        IbT  ^^''       ^^        dx 


(«') 


nfa:" 


(.-2,)(2-|) +  (2.-41 - 


Substituting  the  vahie  of  ~  from  [a'),  we  have 

(Vj  _  {x  -  2//)  (-  3//)  -t-  3r(2.r  -  //) 
dx''  ~  {x  -  tjiY 

-        Cf-27/r'      -(x--2.yf 

EXERCISES. 

Find  by  the  above  method  the  first  two  or  three  derivatives 
of  V  with  respect  to  x,  y  or  z,  from  the  following  equations: 


I.   zv 


=  a(v  -  z).     An,.  ^-  =  j^,. 


2.  V  y  +  vy   =  a. 

3.  y'  +  z^.T  +  //'  =  b. 

4.  r(^<  —  .r)'  +  v''{b  -{-  x)  =  c. 

5.  log  {v  4-  z)  +  log  (y  -  2;)  =  c. 

6.  sin  7/;?;  —  sin  y^?/  =  h. 

7.  ?'(1  —  ((  cos  z)  =  h. 

8.  If  ?<^  —  (3  sin  ?^  =  (J,  show  that 


ded(j       (1  —  c  cos  u)*' 


(«') 


fivatives 

tions: 


) 


DKltlVATIVKS  OF  HIOUKlt  OltDKltS. 


81 


41).  Leibmtz'h  Thkohem.  ToJUkI  the  sucvcHsice  dcruut- 
tives  of  a  product  in  terms  of  the  successive  derivatives  of  Us 
factors. 

Let  /</'  '  p  be  the  product  of  two  fuiictious  of  x.  By  suc- 
cossive  dilfcrcntiiition  we  find 

dp         dv    ,     da 
dx         ax        dx 

jlu  dv  .    d^u 


dx^~  ''W'^    -'-  -'-'^  -' 


1      /v  v 

dx*  dx  dx       dx^    ' 


d*p  _     d'v        (lu  d''v         (l^n  dv      d^u 
(W  ~  '^'d^  '^    '(U'dx^^    d?  (ix  "^  dx'^' 

So  far,  the  coeffi(?icnts  in  the  second  member  are  those  in 
the  development  of  the  powers  of  a  binomial.  To  prove  that 
this  is  true  for  the  successive  derivatives  of  every  order,  we 
note  that  each  coefficient  in  aTiy  one  equation  is  the  sum  of 
the  corresponding  coefficient  plus  the  one  to  the  left  of  it  in 
the  equation  preceding.     Now,  let  us  have  for  any  value  of  n 


d^'p         d''v   ,      du  r/**-'y   ,     ^ 
-~-  —  U-J--  -\-  n-. — T- — .  +  etc. ; 
f/.6"         dx""    '     dx  dx"-^   '  * 


(^0 


the  successive  coefficients  being 

1;     n;     ^Ij;     ^|j ;     etc.  (Comp.  §  6.) 

Then,  in  the  derivative  of  next  higher  order  the  coefficients 
will  be 


1;     n  +  U     [l)  + 
and,  in  general, 

(3  + 


n    or 


m^ 


.s-  1 


u-\-l 


d^'  +  ^p 


That  is,  -j-^TT'i  ^^  formed  from  (n)  by  writing  n  -\- 1  for  n. 

Hence,  if  the  rule  is  true  for  n,  it  is  also  true  for  ?i-\-  1.  But 
it  is  true  for  ?i  =  3;  .  • .  for  71  =  4,  etc.,  indefinitely. 


i''  I 
^1 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


'/ 
^ 


// 


(/  ;^^^ 


/- 


f/. 


1.0 


I.I 


1.25 


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82 


THE  DIFFERENTIAL  CALCULUS. 


50.  Successive  Derivatives  with  respect  to  Several  Equi- 
crescent  Variables.  Studying  the  process  of  §  45,  it  will  be 
seen  that  we  supposed  the  successive  increments  of  the  inde- 
pendent variable  to  be  equal  to  each  other,  and  to  remain 
equal  as  they  became  infinitesimal,  while  the  increments  of 
the  functions  were  taken  as  variable.  This  supposition  has 
been  carried  all  through  the  subsequent  articles. 

Def.  A  variable  whose  successive  increments  are  supposed 
equal  is  called  an  oquicrescent  variable. 

We  are  now  to  consider  the  case  of  a  function  of  several 
equicrescent  variables. 

If  we  have  a  function  of  two  variables, 

the  derivative  of  this   function  with  respect  to  x  will,  in 
general,  be  a  function  of  x  and  y.     Let  us  write 


du 

dx 


7  =  (t>x{^,  y)- 


Now,  we  may  differentiate  this  equation  with  respect  to  y 
with  a  result  of  the  form 


div 
dx 
~dy 


=  ^x,A-'^,y)' 


Using  a  noteUin  similar  to  that  already  adopted,  we  rep- 
resent the  first  member  of  this  equation  in  the  form 

d'^u 
dxdy 

In  the  /^-notation  this  is  written 

In  either  notation  it  is  called  ^Uhe  second  derivative  of 
u  with  respect  to  x  and  y." 
As  an  example:  If  we  differentiate  the  function 

i{>  =  y^  sin  {pxx  —  ny)  {a) 


DERIVATIVES  OF  HIGHER  ORDERS. 


83 


(a) 


with  respect  to  x,  and   then  differentiate    the  result  with 
respect  to  y,  we  have 

BxU  =  —  =  my  cos  {tnx  —  ny) ; 


i>'x..?* 


^'w 


—  2my  cos  (//ia;  —  wy)  +  mny^  sin  (ma;  —  uy). 

51.  We  now  have  the  following  fundamental  theorem: 

d^u         (I'll 
dxdy  ~  dydx^ 

or,  in  words. 

The  second  derivative  of  a  function  luith  reH2)ect  to  two 
equicrescent  variables  is  the  same  whether  we  differentiate  in 
one  order  or  the  other. 

Let  u  =  <f){x,  y)  be  the  given  function.     Assigning  to  x 

the  increment  Ax,  we  have 

^u  _  (f){x  +  Ac,  y)  -  <p{x,  y) 

Jx  ~  Jx         ~  '  ^   ' 

In  this  equation  assign  to  y  the  increment  /ly,  and  call  J—r- 

Au 
the  corresponding  increment  of  -j-.     Then  the  equation  will 

give 

Ax  Ax  Ax 

Subtracting  (1)  and  diviiling  the  dillerenco  by  Ay,  we 
have 

^^  ^  0(g  -[■  My-\-  Ay)  -  (pjx,  y  +  J//)  -  <p{-r  +  Ax,  y)  ■\-^x,y) 
Ay  A.rAy 

The  second  member  of  this  equation  is  symmetrical  with  re- 
spect to  X  and  y,  and  kg  remains  unchanged  Avlicn  \vc  inter- 
change these  symbols.     Hence  we  have 

.All 


.Alt 

^: 

Ay 


Ay 
Ax 


1 


■  'I 

m 


i 


i  il 


84 


THE  DIFFh;BENTIAL  CALCULUS. 


for  all  values  of  Jx  and  Jy,  and  therefore  for  infinitesimal 
values  of  those  increments.     Thus 


dx 
ly 


d 


du 

dy 

dx  ' 


or 


D\,yU  -  D\^u, 


as  was  to  be  proved. 

As  an  example,  let  us  find  the  second  derivative  of  (a)  in 
the  reverse  order.     We  have 


du 
dy 


=  2/y  sin  {7hx  —  ny)  —  n%f  cos  {inx  —  ny); 


d'n 


--  =  2m y  cos  {mx  —  ny)  +  fnny^  sin  {?)tx  —  ;///); 


dydx 

the  same  value  as  before. 

Corollary.  The  result  of  taking  any  numher  of  succes- 
sive derivatives  of  a  function  of  any  number  of  variables  is 
independent  of  the  order  in  tvhich  tve  perform  the  differentia- 
tions. 

For,  by  repeated  interchanges  of  two  successive  differentia- 
tions, we  can  change  the  whole  set  of  differentiations  from 
one  order  to  any  other  order. 

If  we  have  I  differentiations  with  respect  to  x,  in  with  re- 
spect to  ;?/,  n  with  respect  to  z,  etc.,  and  use  the  i)-notation, 
we  express  the  result  in  the  form 

DiD,-^D.^ ...  0. 

Here  the  symbol  D^  means  D^Dy,  etc.,  m  times. 
In  the  usual  notation  the  same  operation  is  expressed  in 
the  form 

dx^dy"'dz'' .  .  .* 

The  corollary  asserts  that,  using  the  jD-notation,  we  may 
permute  at  pleasure  the  symbols  DJ,  DJ^,  D^,  etc.,  without 
changing  the  result  of  the  differentiations. 


DERIVATIVES  OF  BIGBEB  ORDERS. 


86 


EXERCISES. 

Verify  the  theorem  D^DyU  =  DyDji  in  the  following  cases: 

I.  u  =  X  sin  y  -{-  y  am  x.  2.  u  =  x^. 

3.  u  —  x\ogy. 

4.  u  =  asm  (x  -\-  y)  —  h  sin  {x  —  y). 

Differentiate  each  of  the  following  functions  once  with  re- 
spect to  z,  twice  with  respect  to  ?/,  and  three  times  with  re- 
spect to  X,  in  two  different  orders,  and  compare  the  results. 


a'  -  z'' 


6.  ax*y^z*. 


7.  a;  sin  ?/  +  y  sin  2;  +  ^  sin  x.    8,  sin  (Ix  +  wy  +  m), 
9.  If  u 


»\» 


show  that 


^{x'  -\-f-\-  z') 

d'ti      d^u      d'u  _ 
~d?  '^~Ty^'^~dz'~    ' 


53.  Notation  for  Powers  of  a  Differential  or  Derivative. 
Such  an  expression  as  dii*  may  be  ambiguous  unless  defined. 
It  may  mean  eith 

Differential  of  square  of  w;  i.e.,  d(u^)'y 

Square  of  differential  of  w,  i.e.,  {dtiy. 


or 


To  avoid  ambiguity,  the  expression  as  it  stands  is  alwayn 
supposed  to  have  the  latter  meaning.  To  express  the  differ- 
ential of  the  square  of  u  we  may  write  either 

d'u*    or    d{ti^), 

of  which  the  first  form  is  the  easier  to  use. 

dU' 
The  square  of  the  derivative  -j-  may  be  written  either 


duV 


ldu\ 
\dxl 


or 


d_^ 
dx*' 


m 


'm\ 


I, 


I 


86 


THE  DIFFERM^TIAL  CALCULUS. 


CHAPTER  VII. 

SPECIAL  CASES  OF  SUCCESSIVE  DERIVATIVES. 

53,  Successive  Derivatives  of  a   Power  of  a  Derivative, 
Let  us  have  to  differentiate  the  deriyatiye 


(duV 
\dx) 


with  respect  to  x. 

In  such  operations  the  Z)-notation  will  be  found  most  con- 
venient. 

Applying  the  rule  for  differentiating  a  square,  the  result  is 


(duV 
\dx) 


,  du 
//• 

^du      dx  _    du  d^ 

dx       ~    dx     dx    ~~    dx  dx** 


or,  in  the  /^-notation, 

Dx(D^uy  =  2D^uD*u, 
In  the  same  way,  we  find 

d'iD^uY        (duY-^d*u        ,^    ,,   ,^, 

d'{D^uY        (duy-^  d*u  ,_,    ,„_,^, 


dx  ^dx'    dx'        ^^  '  ^^^^  ^' 


SPECIAL  CASES  OF  SUCCESSIVE  DERIVATIVES     87 


KXERCISES. 


IVES. 
'ivativec 


ost  con- 
'esult  is 


u; 


Write  the  derivatives  with  respect  to  x  of  the  following  ex- 
pressions, y  being  independent  of  x  when  it  is  written  as  an 
equicrescent  variable: 

-  (I)' 


•  \dxl  • 

•  m- 

■  Wi ' 

dudu 
dx  ~dy 

'3-  (i)'(ir- 

\dx)  \dx  J 
[d^yVldW 

'9-  W)  \w] ' 


(d'uV 

'  W) ' 

(duV  du 
\dx)  dy 
(d^t'XIdW 

'^*   \dil  W'l  ' 
fd'tiY 

7-   M  • 


(duV 
^-  \d-yl  • 

^'  yKdyi  • 

(  dHi  y 
^'  \dxdyl' 


du  dv      du  dv 
dx  dy      dy  dx 


...  mm. 

\dxl  \dy/ 
(dwid^'Y 

(d'uY 


54.  Derivatives  of  Functions  of  Functions.  Let  us  have, 
as  in  §  40, 

u=f-(tp),  (1) 

where  tp  is  a  given  function  of  x.  It  is  required  to  find  the 
successive  derivatives  of  u  with  respect  to  x.  We  may  evi- 
dently reach  this  result  by  substituting  in  (1)  for  ^^  its  ex- 
pression in  terms  of  x,  and  then  differentiating  the  result  by 
methods  already  found. 

But  what  we  now  wish  to  do  is  to  find  expressions  for  the 
successive  derivatives  without  making  this  substitution.  To 
do  this,  assign  to  x  the  infinitesimal  increment  dx.  The  re- 
sulting infinitesimal  increment  in  tp  will  be 

df  —  -j-dx. 


ySf 


iM 


'■hA 


i'i 


i 


I  "fl 


I  t 


ll 

1 

J        i 

! 

86 


TllE  DIFFERENTIAL  CALCULUS. 


This,  again,  will  give  u  the  increment 

du  =  g#, 

or,  by  substituting  for  difj  its  value,  and  passing  to  the  de 

rivative, 

du  __  du  dtp 

dx  ~  dip  dx ' 


(2) 


This  is  a  particular  case  of  the  result  already  obtained  in 
§40.  The  second  member  of  (2)  is  a  product  of  two  factors. 
The  first  of  these  factors  is  formed  by  differentiating  a  func- 
tion of  ip  with  respect  to  t/r,  and  is  therefore  another  (derived) 
function  of  tp;  while  the  second  is,  for  the  same  reason,  a 
function  of  x. 

Differentiating  (2)  with  respect  to  x  by  the  rule  for  a  prod- 
uct, we  have* 

,du 


d'u  _  dip    dip      du  d*ip 
dx'  ~  dx  dx        dip  dx' ' 

du  . 


(3) 


Now,  because  -7-7  is  a  function  of  ip,  its  derivative  with  re- 
dify  ^ 

spect  to  X  is  to  be  obtained  in  the  same  way  as  that  of  u. 

If  we  put,  for  the  moment. 


du 


»'  =  ^  ^fW' 


we  have,  as  in  (2), 


du'  _  du'  dip  _  d^u  dip^ 
dx  ~  dip  dx  ~  dip'  dx  * 


.du 
*  The  student  should  note  that  the  expression  —. —  cannot  be  put  in  the 


form 


d^dx 


,  because  the  latter  form  presupposes  that  rf)  and  x  are  two  in- 


dependent variables,  which  is  here  not  the  case.    In  fact,  u  does  not  con- 
tain X  except  in  iff. 


I  It 


SPECIAL  GAaSa  OF  aUCCESaiVE  DERIVATIVEa.     89 


and  hence,  by  substitution  in  (3), 


dx'  "  dfXdxj  '^  dip  dx''* 


w 


which  is  the  required  expression  for  the  second  derivatiye. 

From  this  we  may  form  the  third  and  higher  derivatives 
by  again  applying  the  general  rule  embodied  in  (2),  namely : 

If  if)  is  a  fu7iciion  of  x,  we  find  the  derivative  of  any  func- 
tion, UyOf^hy  differentiating  u  with  respect  to  tj),  and  mul- 
tiplying the  resulting  derivative  ly  ~, 

Prom  the  equation  (4)  we  have 


dx 


,  d*u 
fi' 

u  _  jdipy     dtf)^        (Pu  djl^  <f0 
;'  ~~  \dx  I     dx  dij)*  dx  dx* 


,  du 
d^tp      dij)      du  d*i/) 
dx*     dx        dtp  dx* ' 


By  the  rule  just  given,  we  have 

,  d*u 
fi' 

dip*  __  dhi  dtp^ 
dx     ~  dip*  dx  ' 

y      du 

dip  _  d*u  dip 
dx    ~  dtp*  dx  * 

Hence,  by  substitution  and  aggregation  of  like  terms, 

cPu  __  d*u  fdipy        d*u  d*ip  dtp  ,  du  d*tp 
dx*  ~  7f/Adx)  "^    d^  Wdx'^dip  W' 

Repeating  the  process,  we  shall  find 

d'u  _  d^ufdijA'        dhi  d*ip(dtp\* 
dx*  ""  dtpAdxj  "^    d^'  dx*  \dx) 

d^'uF  d*tpdip    oWyn  ,  dud*t/^ 

"*■  #•  i  dx*~  dx'^    \dx*)  J'^  dip  dx'~'    ^^ 


(5) 


.1 


'I 


ii 


n 


,  I 


I 


! 


■1  ! 


90 


THE  DIFFERENTIAL  CALCULUS. 


Example.    Let  us  take  the  case  of 

u  =  sin  tp, 

i/)  being  any  function  whatever  of  x.    We  may  then  form  the 
successive  derivatives  as  follows: 

dx  ~*  dtp  dx  ~"         '^dx  * 

d*u  .     ,(dtp\\  .d'tp 

dtpd^tp 
dx  dx* 


dx 


d*u  ,/#\*     o  .     , 


.     ,dipd'tb  ,  ,d*il) 

—  sin  tb-j^  — f  +  cos  t^r--Y 

^dx  dx*  ^        ^  dx"" 


dtp  (Ptp 


,fdipy      -    .     .dfpd'tp   ,  , 


dx'' 


EXERCISES. 

Putting  0  =  a  function  of  .r,  find  the  first  three  derivatives 
of  the  following  functions  of  0  with  respect  to  xi 

I,  w  =  cos  0.  2.  M  =  0" 

3,  w  =  0»,  4.  u  =  0*. 

5.  ?*  =  log  0.  6.  w  =  e* . 

7.  w  =  siu  20.  8.  u  =  cos  20. 

55.  Chanffe  of  the  Bquicresceiit  Variable,  Let  the  relation 
between  y  and  a;  be  expressed  in  the  form 

a;  =  0(//),  '        (1) 

and  let  it  be  required  to  find  the  successive  derivatives  of  y 
with  respect  to  x,  regarding  the  latter  as  the  equicrescent. 
We  may  do  this  by  solving  (1)  with  respect  to  y,  and  then 
differentiating  with  respect  to  x  in  the  usual  way. 

But  the  method  of  the  last  article  will  enable  us  to  express 
the  required  successive  derivatives  of  y  with  respect  to  x  in 
terms  of  those  of  x  with  respect  to  y,  which  we  can  obtain 


F^PECIAL  OASES  OF  SUCCESSIVE  DERIVATIVES.     91 

from  (1).     By  differentiating  (1)  as  often  as  we  please,  we 
have  results  of  the  form 


Dy*x  =  <p'''y  =  x"\  ) 


(2) 


etc.  etc. 

a;',  x"f  a;'",  etc.,  thus  representing  functions  of  y. 
From  §  37,  Cor.,  we  have 

dx       Djx      X** 


(3) 


To  obtain  the  second  derivative,  we  have  to  differentiate  a;', 
a  function  of  ?/,  with  respect  to  x  (§  54).     Thus 


From  (2), 


d^y  _       _1    f7«'  dy^ 
dx*  "^       x'^'lLy   dx' 

dx*  _  (Vx  _    f, 
dy  ~  ay''~^  ' 


From  this  equation  and  (3)  we  have 

dx*  ""      «"  "■       Jdxy' 

\dy) 
Differentiating  again,  we  find 

rfV  _  (^  d^  _  i.  ^\  dy 
dx*  ""  \  a;'*    <?t/       x'*  dv  I  dx 


(4) 


_  3a;^^'  -  x'x"'  _    XdyV      dy  dy 


\dvV      dv  dv* 


X 


/& 


dxV 
dy) 


The  above  process  may  be  carried  on  to  any  extent.  But 
many  students  will  appreciate  the  following  more  elegar  t 
method  of  obtaining  the  required  derivatives. 

Imagine  that  we  have  solved  the  equation  (1)  so  as  to 
obtain  a  result  in  the  form 

y  =  F{x).  (5) 


H- 


'I 

I 

1 1 


I  ■ 


92 


THE  DIFFERENTIAL  CALCULUS. 


^ 


1i 


If  in  this  equation  wo  substitute  for  x  its  value  (1),  we  shall 
have  a  result  in  the  form 

y  =  ^(0y),  («) 

which,  of  course,  will  really  bo  an  identity. 

But  we  may  still  differentiate  (5)  with  respect  to  y,  regard- 
ing X  aaa  function  of  y  given  by  (1),  by  the  method  of  §§  40 
and  54.    Thus  we  shall  have 

1  =  1  &  (8«*'^-«-) 

d*i/  _  (PyfdxV  d^y  ^  dx  dy  d*x 
dy*  ~"  dx'\dy  I  dx^  dy^  dy  dx  dy*" 
etc.        etc.  etc.  etc. 

But  from  the   identity  (6)  y  =  y,  which  is  obtained  from 
(5),  we  have 

^-l-    ?l-0-    ^'i^-O-    etc 

Therefore,  substituting  for  the  derivatives  of  x  with  respect 
toy  the  expressions  «',  jc",  etc.,  in  (4),  we  have  the  equations 

x'^^--V 
^  dx-^' 


^n  ^   ,  ^ndy_  _  Q. 
*    dx'^"^  dx~  "' 


X 


n 


d'y 


d'y 


dx 


dx 


,."' 


dy_ 
dx 


=  0; 


^"  g  +  6.'V'  g  +  (4.V"  +  3^")  g  +  .-|-  =  0. 

Solving  these  equations  successively,  we  shall  find  the  values 
of  -J-,  ~,  etc.,  already  obtained. 

56.  Case  of  Two  Variables  Connected  by  a  Third,    The 
case  is  still  to  be  considered  in  which  the  relation  between  x 


SPmCIAL  CA8E8  OF  SUCCESSIVE  DERIVATIVES.     93 


I  the  values 


and  y  is  expressed  in  the  form 

y  =  0.(w);    ^  =  <f>M'  (1) 

From  those  equations  it  is  required  to  find  the  successive 
derivative  of  y  with  respect  to  x. 
The  first  derivative  is  given  by  the  equation 

dy_ 
dy_  _du^_  DuV 
dx  "  dx  "  Dyfl 

du  • 

From  the  manner  in  which  the  second  member  of  this  equa- 
tion is  formed,  it  is  an  explicit  function  of  u  alone.  Hence 
(§  54)  we  obtain  its  derivative  with  respect  to  x  by  taking  its 

derivative  Tvith  respect  to  u,  and  multiplying  by  -r-.    Thus 

dx  d*y       dy   d'x 
d^y  __  du  du^ 


da^ 


dx\' 


Idx 
'  \du 


du  du*    du 

'  dx 


dx   d*y      dy   d*x 
__  du  du*      du  du* 

[du  I 

This,  again,  being  a  function  of  u,  further  derivatives  with 
respect  to  x  may  be  obtained  by  a  repetition  of  the  process. 

EXERCISES. 

Find  the  second  derivative  of  x  with  respect  to  y,  and  also 
of  y  with  respect  to  x,  when  the  relation  gt  x  aud  y  is  given 
by  the  following  equations: 

I.  a;  =  a  cos  u;  y  =  h  sin  u, 

z,  X  =  a  cos  2u;  y  =  h  sinu, 

3,  a;  =  rt  cos  2u;  y  =  J(cos  u  —  sin  w),  "  j 

4,  x  =  u  —  eBinu;  y  =  u-\-6sinu. 


n 


I  , 

1  ■■ ; 
'I .  ■  1 

'hi; 

1 14 


5.  x^^ 


u. 


y  =  ue^. 


ill 


1 1, 


94 


THE  DIFFERENTIAL  CALCULUS. 


I  i 


6.  Show  that  if 

«  ,,         d*u  3  sin  u 

y  =  e"  cos  u,    then     -r-,-  =  -^. ; ;-. 

ay        e*'*(cos  u  —  sm  u) 

'/.  Show  that  the  wth  derivative  of  »"  +  «^"''*  +  ^a;""*  is 
n\,  n  being  a  positive  integer  >  1. 

8.  Show  that 

9.  Show  that  if  v  =  u"*,  then 

i»  =  nu^'-'D^'u  -f-  3w(w  -  l)tc''-W^uDJ'u 

+  ?^(7^  -  1)  (?^  -  2)u''-\D^u)\ 

10.  If  7/  =  a  cos  wia;  +  h  sin  wa;,  show  that 

DJu  +  w'w  =  0. 

Then,  by  successively  differentiating  this  result,  show  that, 
wiaatever  the  integer  w, 

11.  If  «^  =  e*  COS  ic  and  v  =  e*  sin  a;,  then 

D^'tc  =  —  2v    and     D^^v  =  9,2^. 
Also,  I>a,V  4-  4v  =  0; 

12.  If       w  =  e"*  CDS  ma;    and    v  =  e"*  sin  yna;, 

show  that  the  successive  derivatives  of  u  and  v  may  always  be 
reduced  to  the  form 

DJu  =  Afic  —  BiV;    DJv  =  AiV  -{•  BtU,  (a) 

where  A  and  B  are  functions  of  v%  and  w.  Also,  find  the 
values  of  ^,,  A^,  B^  and  5„  and  sho\^  by  differentiating  (a) 
that 

At  +  i  =  A,At-B,Bt;    B,^,^  B,A,-{- A.B^, 


f 


DEVELOPMENTS  IN  SElilES. 


96 


k»* 


n  uy 

4- Jaj^'Ms 


i 


^u)\ 


IL^-\D^U)\ 


;,  show  that. 


ay  always  be 

(II,  (a) 

Iso,  find  the 
entiating  (a) 


I 


CHAPTER  VIM. 
DEVELOPMENTS  IN  SERIES. 

5*7.  A  series  is  a  succession  of  terms  all  of  whose  values 
are  determined  by  any  one  rule. 

A  series  is  called 

Finite  when  the  number  of  its  terms  is  limited; 

Infinite  when  the  number  of  its  terms  has  no  limit. 

The  sum  of  a  finite  series  is  the  sum  of  all  its  terms. 

The  sum  of  an  infinite  series  is  the  limit  (if  any)  which  the 
sum  of  its  terms  approaches  as  the  number  of  terms  added  to- 
gether is  increased  without  limit. 

When  such  a  limit  exists,  the  series  is  called  convergent. 

When  it  does  not  exist,  the  series  is  called  divergent. 

To  develop  a  function  means  to  find  a  series  the  limit  of 
whose  sum,  if  convergent,  shall  be  equal  to  the  function. 

We  may  designate  a  series  in  the  most  general  way,  in  the 
form 

«*,  +  ^^  +  W,  +  .  .  .   +  «n  +  W„  +  i  +  .  .  .  , 

the  nth.  terms  being  called  w„. 

58.  Convergence  and  Divergence  of  Series.  No  universal 
criterion  has  been  found  for  determining  whether  any  given 
series  is  convergent  or  divergent.  There  are,  however,  a  great 
number  of  criteria  applicable  to  a  wide  range  of  cases.  Of 
these'  we  mention  the  simplest. 

I.  A  series  cannot  he  convergent  unless,  as  n  lecomes  in- 
finite f  the  nth  term  approaches  zero  as  its  limit. 

For  if,  in  such  case,  the  limit  of  the  terms  is  a  finite 
quantity  oc,  then  each  new  term  which  we  add  will  always 


I  1 


:" 


m 


06 


THE  DIFFERENTIAL  CALCULUS. 


change  the  sum  of  the  series  by  at  least  or,  and  so  that  sum 
cannot  approach  a  limit. 

As  an  example,  the  sum  of  the  series 

1  —  1  +  1  —1  +  1  —  1,  etc.,  ad  infinitum, 
will  continually  change  from  +  1  to  0,  and  so  can  approach 
no  limit,  and  so  is  divergent,  by  definition. 

II.  A  series  all  of  whose  terms  are  positive  is  divergent 
unless  nUn  =  0  when  n^^  ao.    < 

To  prove  this,  we  have  first  to  show  that  the  harmonic 

series 

i  +  |  +  i  +  -^+  etc.,  ad  infinitum, 

is  divergent.     To  do  this  we  divide  the  terms  of  the  series, 

after  ihe  first,  into  groups,  the  first  group  being  the  2  terms 

^  +  :|-,  the  second  group  the  following  4  terms,  the  third 

group  the  8  terms  next  following,  and,  in  general,  the  nih. 

group  the  2"  terms  following  the  last  preceding  group.     Wo 

shall  then  have  an  infinite  number  of  groups,  each  greater 

than  ^. 

Now,  if,  for  all  the  terms  of  the  series  after  the  nth,  we 

have 

nUn  >  a  {a  being  any  finite  quantity), 

then  Un  >  -, 

n 

1 


(I  1 

and  Un  +  Um+i  +...>«-  H rr  +       •  o 

'        ^  \w      w  + 1      m  +  2 


+ 


•    •    •  /< 


Because  the  last  factor  of  the  second  member  of  this  equa- 
tion increases  to  infinity,  so  does  its  product  by  a,  which 
proves  the  theorem. 

III.  If  the  terms  of  a  series  are  alternately  positive  and  nega^ 
tive,  continually  diminish,  and  approach  zero  as  a  limit, 
then  the  series  is  convergent* 

Let  the  series  be 

w,  —  w,  +  «*,  —  w*  +  w.  — 
Then,  by  hypothesis, 

w,  >  w,  >  w,  >  w,  >  . 


*  •  •  • 


DEVELOPMENTS  IN  SEIilES. 


97 


the  series, 
he  2  terms 
the  third 
il,  the  7ith. 
roup.  We 
ich  greater 

[le  nihy  we 


+  ...). 

this  eqiia- 
aj  which 


e  and  nega^ 
IS  a  limit f 


Let  us  put  Sn  for  the  sum  of  the  first  n  terms  of  the  series, 
71  being  any  even  integer,  and  S  for  the  limit  of  the  sum,  if 
any  there  be.  Then  this  limit  may  be  expressed  in  either  of 
the  forms 

and 

S=  Sn  +  l  —  {Un  +  9  —  Un  +  s)  —  (Wn  +  4  "  ^n  +  b)  —  •   •  •  • 

Since  all  the  differences  in  the  parentheses  are  positive,  by 
hypothesis  it  follows  that,  how  many  terms  soever  we  take, 
the  sum  will  always  be  greater  than  S^  and  less  than  S^+i* 
The  difference  of  these  quantities  ib  tin ^ i,  which,  by  hypothe- 
sis, approacliea  zero  as  a  limit.  Since  the  two  quantities  /S'„ 
and  Sn+i  .ipproach  indefinitely  near  each  other  from  opposite 
directions,  they  must  each  approach  a  limit  S  contained  be- 
tween them. 

Graphically  the  demonstration  may  be  shown  to  the  eye 
thus;    Let  the  line  OS^  represent  the  sum  S^,  when  «  =  6, 

O^  Se      Ss    Sia— S Sn  S9    Sr 


I     I    t- 1    I    I 


Fio.  11. 

or  any  other  even  number;  OS^  the  sum  S^,  etc.  Then  every 
succeeding  even  sum  is  greater  than  that  preceding,  and 
every  succeeding  odd  sum  is  less  than  that  preceding,  while 
the  two  approach  each  other  indefinitely.  Hence  there  must 
be  some  limit  S  which  both  approach. 
An  example  of  such  a  series  is 


3^  5 
of  which  the  7ith  term  is  — 


y  +  --etc., 


(-1)« 


We  shall  hereafter  see 


2n  -  1 

that  the  limit  of  the  sum  of  this  series  is  Itt.     If  we  divide 

the  terms  into  pairs  whose  sums  are  negative,  the  series  may 

be  written 

2 


3'5       7-9 


11  13 


etc. 


i 


m 


H 


rf 


' 


3 


Mi 


;< 


!( n      i 


I  M 


]^  I*    t 


98 


2UIB  DIFFERENTIAL  CALCULUS. 


Pairing  the  terms  so  that  the  sum  of  each  pair  shall  be  posi- 
tive, the  series  becomes 


2  2  3  2 

3  +F7  +  901  +  1305  +  ^^''' 

We  may  show  by  the  preceding  demonstration  that  these 
series  approach  the  same  limit. 

IV.  If  J  after  a  ccrtai7i  finite  mcmler  of  terms ,  the  ratio  of 
tivo  consecuiivG  terms  of  a  series  is  continually  less  than  a  cer- 
tain qnaittity  a,  luhich  is  itself  less  than  unity j  then  the  series 
is  convergent. 

Let  the  nth  term  be  that  after  which  the  ratio  is  less  than 
a.     We  then  have 

«*n+2  <  ««*«  +  !  <  «X; 

•  •  t  •  •  • 

Taking  the  sum  of  the  members  of  these  inequalities,  we 
have 

But  a  -{-  a*  ■{-  a*  -\-  »  ,  ,  is  B>n  infinite  geometrical  progres- 


sion whose  limit  when  a  <  1  is 


a 


a: 


,  a  finite  quantity. 


Hence,  putting  *S^  for  the  limit  of  the  sum  of  the  given 
series,  we  have 


S<S^  + 


a 


1  -  «"»• 


The  second  member  of  this  inequality  being  a  finite 
quantity  which  S  can  never  reach,  S  must  have  some  limit 
less  than  that  quantity. 

As  an  example,  let  us  take  the  exponential  series 


DEVELOPMENTS  IN  SERIES. 


99 


a  that  these 


[)  is  less  than 


The  ratio  of  the  in  +  l)st  to  the  nih  term  is  -.     This 

n 

ratio  becomes  less  than  unity  when  n  >  x,  and  it  approaches 

zero  as  a  limit.     Hence  the  series  is  convergent  for  all  values 

of  X. 

Corollary.    A  series 

proceeding  according  to  the  jjowers  of  a  vai'iable,  x,  is  conver- 
gent when  X  <  1,  provided  that  the  coefficients  a^  do  not  in- 
crease indefinitely. 

Remakes. — (1)  Note  that,  in  applying  the  preceding  rule,  it  does  not 
suffice  to  show  that  the  ratio  of  two  consecutive  terms  is  itself  always 
less  than  unity.  This  is  the  case  in  the  harmonic  series,  but  the  series  is 
nevertheless  divergent.     The  limit  of  the  ratio  must  be  less  than  unity. 

(2)  If  the  limit  of  the  ratio  in  question  is  greater  than  imity,  the  series 
is  of  course  divergent.  Hence  the  only  case  in  which  Rule  IV.  leaves 
a  doubt  is  that  in  which  the  ratio,  being  less  than  unity,  approaches 
unity  as  a  limit.    But  most  of  the  series  met  with  come  into  this  class. 

(3)  The  sura  of  a  limited  number  of  terms  of  a  series  gives  no  certain 
indication  of  its  convergence  or  divergence.  If  we  should  compute  the 
successive  terms  in  the  development  of  <?  -  loo  we  should  soon  find  our- 
selves dealing  with  numbers  having  thirty  digits  to  the  left  of  the  deci- 
mal-point, and  still  increasing.  But  we  know  that  if  we  should  continue 
the  computation  far  enough,  say  to  1000  terms,  the  positive  and  negative 
terms  would  so  cancel  each  other  that  in  writing  the  algebraic  sum  we 
should  have  42  zeros  to  the  right  of  the  decimal-point. 

On  the  other  hand,  if  the  whole  human  race,  since  the  beginning  of  his- 
tory, had  occupied  itself  solely  in  computing  the  terms  of  the  harmonic 
series,  the  sum  it  would  have  obtained  up  to  the  present  time  would  have 
been  less  than  44.  For  1000  million  of  people  writing  5000  terms  a  day 
for  2  million  of  days  would  have  written  only  10^"  terms.  It  is  a  thco  -  n 
of  the  harmonic  series,  which  we  need  not  stop  to  demonstrate,  that 


^=2+3+4-  +  -- 


-| —  <  Nap.  log  n. 


But       Nap.  log  10"»  =  S25^:j2giO^'  ^  _19_ 
and  yet  the  limit  of  the  sum  of  the  scries  is  infinite. 


=  43.78, 


i  -'111 


u- 


:'i 


^'i, 


M 

m 


il 


1 


I  '! 


y-  -( 

T     -■ " 

1 

i 

1  ^    ' 

i 

i   f  ' 

i 

(               ; 

i 

i 

[              ; 

i 

)    ! 

1                          J 

J 

1 

! 

1 

1 

! 

!  j 

i , 

100 


THE  DIFFERENTIAL  CALCULUS. 


59.  Maclmirin's  Theorem.  This  theorem  gives  a  method 
of  deyeloping  any  function  of  a  Tariablo  ia  a  series  proceod- 
ing  according  to  the  ascending  powers  of  that  variable. 

If  X  represents  the  variable,  and  0  the  function,  the  series 
to  be  investigated  may  be  written  in  the  form 

cp{x)  =  ^0  +  A,x  +  A,x'  +  A,x'  +  .  .  .  ;  (1) 

the  series  continuing  to  infinity  unless  0  is  an  entire  func- 
tion, in  which  case  the  two  members  are  identical. 

Whether  the  development  (1)  is  or  is  not  possible  depends 
upon  the  form  of  tlie  function  0.  Most  functions  admit  of 
being  so  developed;  but  special  cases  may  arise  in  which  the 
development  is  not  possible.  Moreover,  the  development  will 
be  illusory  unless  the  series  (1)  is  convergent.  Commonly  this 
series  will  b^  convergent  for  values  of  x  below  a  certain  mag- 
nitude, often  unity,  and  divergent  for  values  above  that  mag- 
nitude. "What  we  shall  now  do  Is  to  iissume  the  development 
possible,  and  show  how  the  values  of  the  coefficients  A  may  be 
found. 

Let  us  form  the  successive  derivatives  of  the  equation  (1). 
We  then  have 

<t){x)  =  ^0  +  A^x  +  A^x"  4-  etc.; 
d(t> 


dx 
d'cfy 


=     cf)\x)  =  ^,  +  ^A^x  +  ZA,x'  +  .  .  .  ; 


dx 

ay 

dx' 


P=  0"(a:)  =  l-2^,  +  2-3^3rc  +  3-4^,a;'  + 


=  0'"(a;)  =  l-2 


dy 
dx"" 


=  0<'»)(a:)  =  1-2 


3^, +  2-3-4^,2;4-  .  .  .  ; 


3  •  4  .  ,  .  7iAn  +  etc. 


By  hypothesis  these  equations  are  true  for  all  values  of  x 
small  enough  to  rei.  ler  the  series  convergent.  Let  us  then 
put  X  =  0  in  all  of  them.     We  then  have 


DEVELOPMENTS  IN  SERIES. 


101 


}  a  method 
5S  proce«,d- 

,  the  serios 

;         (1) 

ntirc  func- 

ble  depends 
s  admit  of 
1  which  the 
Dpment  will 
nmonly  this 
ertain  mag- 
e  that  mag- 
levelopment 
ts  A  may  be 

^uation  (1). 


"T  •  •  •  5 


0(0)  =  ^„; 

0'(O)  =  A,) 


.*.  A^ 
•  *  •  ^j 


0(0). 
0'(O). 


0"(O)  =  l-2^.;        .•.^.  =  ji-^0"(O). 
0'"(O)  =  1  2-3^.;    .-.  ^,  =  p^0"'(O). 


0(«)(O)  =  n\An\ 


1^ 


.•.^„  =  -^0(»)(O). 


By  substituting  these  values  in  (1)  we  shall  have  the  re- 
quired development.  Noticing  that  the  symbolic  forms  0'(O), 
0"(O),  etc.,  mean  the  values  which  the  successive  derivatives 
take  when  we  put  x  =  0  after  differentiation,  we  see  that  the 
coefficients  are  obtained  by  the  following  rule  : 

Form  the  successive  derivatives  of  the  given  function. 

After  the  aerivatives  are  formed,  suppose  the  variable  to  he 
zero  in  the  original  function  and  in  each  derivative. 

Divide  the  quantities  thus  formed,  in  order,  l?yl',  1;  1*2; 
1'2*3,  etc.,  the  divisor  of  the  nth  derivative  leing  n\ 

The  quotients  will  he  the  coefficients  of  the  powers  of  the 
variahls  in  the  development,  commencing  with  the  zero  power, 
1  or  absolute  term, 

EXAMPLES    AND    EXERCISES. 

I.  To  develop  (a  +  ^)"  ^  **  ^^  powers  of  x.    We  have 

•  •  -A.  ^^  a  , 


u  =  {a  ■-{-  xY; 

lu 

dx 

dx 


.•.  -4,  =  na^-K 


^'^  '      f        i\  /     I     \n_2  M        n{n  —  1)  „    , 


m 

km 

'"nil 


t}  i 


At 


1,1  .1 


t 


lvalues  of  x 
Let  us  then 


'dx^ 


=  n(n  —  l),,,{n  —  s-\-l)(a-\-  a;)"-". 


'I    4 


;    ! 


1    f 


102  TITE  DIFFERENTIAL  CALCULUS. 

Thus  the  development  is 
{a  +  xy  =  a"  +  m--'x  +  (|)a"- V  +  (|)«— V  -j-  .  .  .  , 

which  is  the  binomial  theorem. 

2.  Develop  («  —  a:)"  in  the  same  way. 


3.  Develop  log  (1  +  x). 
Here  we  shall  have 


=  (i  +  a^)-»; 


du  _      1 
fl?aj  ""  1  +  a; 

I?  =  1-3(1  +  ^)-'; 


etc. 


etc. 


Noticing  that  log  1  =  0,  we  shall  find 

log  {l  +  x)=x-ix'-\-^x'-ix*  + 

4.  Develop  log  (1  —  x). 

5.  Develop  cos  x  and  sin  x. 

The  successive  derivatives  of  sin  x  are  cos  x,  —  sin  x,  —  cos  x,  sin  «, 
etc.    By  putting  x—0,  these  become  1,  0,  —  1,  0,  1,  0,  etc.    Thus  we 


find 


/p'i  /}A  /i^ 

3?^         2!^         3j' 

8in«  =  aj-3j+---  +  .... 


6.  Develop  e*,  where  e  is  the  Naperian  base. 


X'    .    X' 


Ans,  6»'  =  l+.T  +  ^  +  ^^  + 


7.  Develop  e"'". 

8.  Show  that 


•    •    •    • 


-111          .   i^^^SC'V   ,   (xlosa)'  , 
fl'"  =  1  +  aj  log  ^,  +  ^    ,  °    ^  4-  ^  ,.°.^^  + 


1-2 


1-2-3 


9.  Deduce  e«»°'  =  i4.a;  +  ^_^4. 

2        4! 


DEVELOPMENrS  IN  8ERIE8. 


103 


10.  Develop  sin  (a  +  x)  and  cos  (n  -^x)  and  thonco,  by  com- 
paring with  the  results  '^f  Ex.  5,  prove  the  formulaB  for  the 
sine  and  cosine  of  the  sum  of  two  arcs.     Find  first 

-a 

^j    +    .    .)    4-C0S«    (^-gj 

11.  Develop  (1  -f  <'*)'*  and  show  that  the  result  may  bo  re- 
duced to  the  form 

n      .  n^  -\-  n  x*       n^  -f  3n'  x^ 


X'  X 

sin  (tt  -f-  x)  —  sin  «  (1  —  n-r  +  •  •)  +  cos  a  {x  —  ^+  .  .). 


2 


r^2^^     2'      1.2^        2-        31  ^' 'I* 
1 2.  Develop  e*  sin  x  ana  c*  cos  x  and  deduce  the  results 


x*       ^x'        .  .'t'       „  a:" 


.«sin^  =  :.  +  2^+23j-4:gj-8gj   -... 


e"  cos  a;  =  1  +  a;  -2^  --  4?-r  -  4^  + 


2!       '4!         51 

13.  Develop  cos'  x. 

Begin  by  expressing  co8^  x  in  the  form  i  cos  Sa;  4-  f  cos  x. 

1 4.  Develop  tan  ^""  %. 

This  case  affords  us  an  example  of  how  the  process  of  de- 
velopment may  often  be  greatly  abbreviated.  It  has  been 
shown  that 

f?-tan(-^)a;  1 


_  -— — -  =z  I  —  X*  -{- X*  —  X*  4-  etc. 

dx  1  +  «•  ' 

Now  assume 

tan^~%'  =  A  -\-  A^x  -\-  A^x'  -\-  etc. 

This  gives 

<?*tan^-% 


(a) 


dx 


=  A^-{-  2A,x  +  SA,x^  +  etc. 


W 


Comparing  {a)  and  (i),  we  have 

J,  =  l;    A,  =  -};    A^  =  \;    A,  =  -  \;    etc. 
and  A^  =  A^  =  A^.  .  .  =  0. 

The  value  of  ^^  is  evidently  zero.     Hence 

tan<-»)a;  -=  x  -  ix'  +  ^x'  —  ^x'  +  etc. 


i^) 


lit  ;  I* 


t 


ri! 


ill 


'*! 


f 
t 

1 


I.  I 


Mill 


• 


101 


TUe  DIFFERENTIAL  CALCULUS. 


15.  Develop  sin<~*^a;. 

c?-8in<~'' 


Since 


dx 


•■  (1  -  «•)- », 


we  may  develop  the  derivative  and  proceed  as  in  the  last  ex- 
ample.    We  shall  thus  find 


.  ,  ,,        X    ,  1   X*      1-3  X*  ,  l-3'6  x*  ,     , 
8m<%  =  -  +  ^.3-  +  ^^.-+^^^+etc. 


60.  Ratio  of  the  Circumference  of  a  Circle  to  its  Diameter, 
The  preceding  development  of  tan^~*^a;  affords  a  method  of 
computing  the  number  n  with  great  ease.  The  series  (c) 
could  be  used  for  this  purpose,  but  the  convergence  would  be 
very  slow.  Series  converging  more  rapidly  may  be  obtained 
by  the  following  device: 

Let  a,  a\  a",  etc.,  be  several  arcs  whose  sum  is  45°  =  \it. 
We  then  have 

tan  (a  +  a'  4-  a"  4-  etc.)  =  1. 

Let  t,  V,  /",  etc.,  be  the  tangents  of  the  arcs  a,  a*,  a'\  etc. 
If  there  are  but  two  arcs,  a  and  or',  we  then  have,  by  the 
addition  theorem  for  tangents, 

1±^  =  1;    or    ,?  +  i{'  =  l-«'. 

If  there  are  three  arcs,  a,  a',  and  a*\  we  replace  t*  by 
V  + 1" 


in  the  last  expression,  and  thus  get 


1  -  tH" 

t^t'  +  f'  -  tft"  =  !-«'-  ft"  -  tt". 

We  now  have  to  find  fractional  values  of  t,  t'  and  t"  of  the 

form  — ,  m  being  an  integer,  which  will  satisfy  one  of  these 

equations.  Unity  is  chosen  as  the  numerator  because  the 
powers  of  the  fraction  are  then  more  easily  computed.  The 
simplest  fractions  which  satisfy  the  last  equation  are 

111 
^-2'    ^  --5'    ^    -8* 


DEVELOPMENTS  IN  8ERTKS. 


106 


We  then  have,  from  tho  development  of  tan  <~*^  t,  etc., 


the  last  ex- 


ft         a  •  •> »  ~  ft  •  9. » 


2       3-2 


a'  =  =^- 


6-3' 


•  •  •  > 


3-5"   '   5-6 


•  •  •  > 


[-etc. 

i7s  Diameter, 
a  method  of 
he  series  (c) 
snce  would  be 
ybe  obtained 

1  is  45°  =  \^' 


a,  «',  «">  e*c- 
,n  have,  by  the 


a"  = 1-  -— 


^'  and  ^"  of  the 
If  y  one  of  these 


8       3-8 


6-8' 


—  jt  =  a  -{■  a'  -\-  a". 

These  series  were  used  by  Daso  in  computing  n  to  200 
decimals. 

A  combination  yet  more  rapid  in  ordinary  use  is  found  by 
determining  oc  and  a'  by  the  conditions 


tan  a  =  -' 
5 


^a  —  a'  =  -r  Tt, 
4 


We  then  have 


tan  2ar  = 
tan  4ar  = 


120, 
119' 


and  because  «'  =  4^  —  \7t  =  4a'  —  45°,  we  have 

,  _  tan  4a'  —  1  _    1 
^^"""^  ~  tan  4«  +1  ~  239- 

Hence  we  may  compute  tt  tl:  an  : 


a  =  -~ 


«'  = 


1     JL  .  J L.  4. 

3-5''"^5-5'      7-5^"^*  •  •' 


"T"  K.0QQ6 


239      3-239"   '  5*239' 
7f  =  4.a  —  a'. 

Ten  or  eleven  terms  of  the  first  series,  with  four  of  the 
[second,  will  give  ;r  to  15  places  of  decimals. 


]'hA 


i 
ii. 


■•I 

;1 


til 


*. 


1,  •; 


» 


106 


TUE  DIFFERENTIAL  CALCULUS 


61.  In  developing  functions  by  Maclaurin's  theorem  wo 
may  often  bo  abb  to  express  the  derivatives  of  a  certain  order 
as  functions  of  those  of  a  lower  order.  Tlic  process  of  find- 
ing the  higlier  derivatives  may  then  be  abbreviated  by  retain- 
ing the  derivatives  of  lower  orders  in  a  symbolic  form,  so  far 
as  possible. 

EXAMPLES. 

I.  Lot  us  develop 

n  =  log  (1  -f-  sin  a:)  E  0(x). 
We  now  have 

cos  X         1  —  sin  X 


<P'(cc)  = 


=  sec  a;  —  tan  x; 


1  -}-  sin  :c  cos  x 

<p"(x)  =  sec  X  tan  x  —  sec"  a;  =  —  sec  x(/)'{x). 

Now,  in  continuing  the  differentiation,  we  use  the  last  of 
tliese  forms  instead  of  the  middle  one.     Thus 

ft>'"[x)  =  —  sec  X  tan  x  (f>'{x)  —  sec  x<p''{x) 
=  —  sec  a;  tan  x  <f>'{x)  +  sec'  x(/)'{x) 
=  -  <P'{x)cfy"{x), 

We  may  now  find  the  successive  derivatives  symbolically. 
Omitting  the  symbol  x  after  0,  we  have 
0»v  _  _  0'0"'  _  0'". 
(ff  =  —  0'0'v  -  30" 0"'; 
0v»  =  —  0'0v  _  40" 0"'  —  30"". 


etc. 

Supposing  a;  =  0, 

0(0)  =  0; 
0'(O)  =  +  1 
0"(O)  =  -  1 
0"'(O)  =  +  1 


etc. 

0'-(O)  =  -  2; 
0v(O)  =  +  5; 
0vi(O)  z=  -16; 
etc.  etc. 


Hence 


log(l  +  sin..)=»:-^-  +  |--i^  +  |j 


x" 
45 


+  ... 


DEVELOPMENTS  IN  SERIES. 


107 


2.  To  develop  u  =  tan  x. 

Let  us  write  the  equation  in  the  implicit  form 

u  cos  a:  —  sin  a;  =  0. 

Then,  by  dififerontiation  and  division  by  cos  x,  we  find 
D,u  =  1  +  w'; 
DJu  =  2uD^u  =  2w  -f  2u*; 
D^'u  =  2uDJ'u  +  2(2)^?0'; 
D^'u  =  2uD^*u  +  SD^uDJu  +  C(/)/?0'' 

Putting  w  =  0,  we  find  the  even  derivatives  to  vanish  and 
the  odd  ones  to  become  1,  2,  IG,  etc.     Hence 

tan  X  =  X  -\-  ^x*  -\-  ^x^  +  .  .  .  . 

3.  To  develop  rt  =  sec  x. 

Differentiating  the  form  n  cos  a:  —  1  =  0,  wo  find 

Da.u  cos  X  —  u  sin  cr  =  0.  (a) 

The  successive  derivatives  of  this  equation  may  each  be 
written  in  the  form 

M  COBX  —  N  bin  a;  =  0.  (h) 

For,  if  we  differentiate  this  equation  with  respect  to  a:,  it 
becomes 

(D^M  -  N)  cos  x-{M-\-  D^N)  sin  x  =  0. 

Hence  the  derivative  of  (h)  may  be  formed  by  putting 

M'  =  D^M  -  JST;     N'  =  M -{■  D,N,  (c) 

and  writing  M'  and  N^  instead  of  M  and  JV  in  the  equation. 
In  (a)  we  have 

M  =  D^u;    ]Sf=  u. 

Then,  by  successive  substitution  in  (c), 

M'  =  DJ'u-  u;  N'  =  2D^u; 

M"  =  D^'u-  SD^u;  N"  =  SDJu  -  u; 

M'"  =  DJu-  GD^'u  +  u;        N'"  =  4:D^'u  -  4:D^u; 
M^"  =  DJ'ti-lODJu  +  5^,.?r;  N^"  =  6D^*u  -  lODJ'u  +  u, 
M"  =  DJu  -  15D^*u  +  IBDJ'u  -  u; 


II 


l>  Hi 

1;    H 


I 


.1 


lOS 


THE  DIFFERENTIAL  CALCULUS. 


i    !; 


i!     11 


'   :i 


ill  I 


I 


When  a:  =  0,  we  have  sin  x  —  0,  cos  x  =  1,  u  =  1,  and 
hence  M=  M'  =  .  .  ,  =0  in  all  the  equations.  Thus  we 
find,  for  x  =  Qj 

B^n  =  M  =  1; 

BJ^u  =  6-1  =  5; 

D^'u  =  75  -  15  4- 1  =  61; 

etc.  etc. ; 

while  the  odd  derivatives  all  vanish.     Hence 

sec  a;  =  1  +  -  a;'  +  ^j  x'  -f  -j-  a;"  + 

63.  Taylor's  Theorem.  Taylor's  theorem  differs  from 
Maclaurin's  only  in  the  form  of  stating  the  problem  and  ex- 
pressing the  solution.     The  problem  is  stated  as  follows: 

Having  assigned  to  a  variable  x  an  increment  h,  it  is  re- 
quired  to  develop  any  function  ofx-{-h  in  powers  ofh. 

Solution,  Let  <p  be  the  function  to  be  developed,  and  let 

us  put 

u=(p{x);         ) 

u'=  <f>(x-\-7i).  I 
Assume 


n'  =  X,  +  Xfi  +  X,/i'  +  A\7i'  +  etc 

where  X„,  X„  etc.,  are  functions  of  x  to  be  determined. 
Then,  by  successive  differentiation,  we  have 

^-  =  uY,  -f  2XJt  +  dXJi'  -f  4:XJi'  4-  etc.; 


(1) 
(2) 


d'u' 


d/t 
d'u' 


J-  =  2X,  +  2-3X3/^  +  3-4Xyi'  +  etc.; 


(3) 


=  l-2-3X3-f  3'3-4X,A  +  etc. 

etc.  etc.  etc. 

We  now  modify  these  equations  by  the  following  lemma: 
If  we  have  a  function  of  the  sum  only  of  several  quantities, 

the  derivatives  of  that  function  with  respect  to  those  quantities 

will  he  equal  to  each  other. 


DEVELOPMENTS  IN  SERIES. 


109 


u  =  1,  and 
B.    Thus  we 


difEers  from 
ablem  and  ex- 
i  follows: 
nt  h,  it  is  re- 
ers  of  h. 
sloped,  and  let 

(1) 


•  •  •  • 


(3) 


termined. 


etc. 


;0«  « 


:1 


(3) 


wing  lemma: 
feral  quantities, 
those  quantities 


For  if  in  f{x  -\-  h)  we  assign  an  increment  Jh  to  x  and  to 
h  separately,  the  results  will  hef{x  -\-h-\-  Ah)  and /(a;  +  JA 
-f-  h),  which  are  equal. 
It  follows  that  we  have 

du'  _  di^' 
dh  ~  dx' 

Now  these  equal  derivatives,  like  u'  itself,  are  functions  of 
X  +  h  alone,  so  the  lemma  may  be  applied  to  as  many  suc- 
cessive derivatives  as  we  please,  giving 

dW      d\t' 


dh' 
dhi' 


dx"' 
d'u' 


dh'       dx' ' 
etc.        etc. 

Now  let  the  derivatives  with  respect  to  x  bo  substituted  for 
those  with  respect  to  h  in  equations  (3),  and  let  us  suppose  h 
to  become  zero  in  equations  (:^)  and  (3).  Then  ii'  and  its  de- 
rivatives will  reduce  to  u  and  its  derivatives,  and  we  shall  get 

du 


x  = 


dx' 


^r- 

1    d'u 
~  I'^dx'' 

• 

1      d'u 
~  l-'Z'^dx'' 

• 

• 

_  1  d^'u 
"  n\  dx'' ' 

Then,  by  substitution  in  (^),  we  shall  have,  for  the  required 

development, 

,   du  h    ,   d'u  h"*    ,   d*u    W      ,     , 

U'  —  u  -\-  -J-  -  -}-   -j-T,  r—-  -f  -y-3  r— t-t;  +  etc. 

dx  1    '    dx'  1-2  '    dx'1'2-3 
This   formula   is    callecj.   Taylor's    Theorem,    after    Brook 
Taylor,  who  first  discovered  it. 


:m 


i  '  ■  I 


I 


I 


VH 


:      i\ 


110 


THE  DIFFERENTIAL  CALCULUS. 


I  " 
I 


(i     ;' 


I'  II! 


EXAMPLES    AND   EXERCISBS. 


I.  Develop  (x  +  A)". 
We  proceed  as  follows  : 


du 
dx 


=  nx^~^\ 


d  u         ,         -.vm.  —  " 
—  =  w(?i-l)ic»    -; 

etc.  etc. 

By  substitution  in  the  general  formula  we  find 


(2;-h/0"  =  a:~  +  p'i;»-^A  +  ^ 


'^*(^*-l)..n-s 


2 


h 


~i       r^2'^"3  -f  •  •  •  • 


2.  Devolop  the  exponential  function  «*"*"''  in  powers  of  U, 
Am.  (f\\  -f  log  a^  +  (log  w)'^^  +  .  .  •  V 


3.  sin  {x  -\-  h). 

5.  sin  {x  —  //). 

7.  log  {x  +  70. 

,      x-\-h 

II.  cos*  {x  -f-  Zf). 
13.  tan(-»>(.c  +  //). 


4.  cos  (:c  +  ^0* 

6.  cos  (:c  —  /i). 

8.  log  (.c  —  A). 

10.  log  cos  X. 

12.  sin'  {x  —  Zi). 

14.  sin^~^^  {x  —  7i). 


15.  Deduce  the  general  formula 


X 


n^) 


-r;i   ~-^'  —  etc. 


(1  +  a:)'    1-2 

16.  Prove,  by  differentiation  and  applying  the  algebraic 
theorem  that  in  two  equal  series  the  coefficients  of  like 
powers  of  the  variables  must  be  equal,  that  if  we  have 

log  («o  +  «i^  +  «a^'  +  •••)  =  ^0  +  ^1^  +K^''  +  '  '  '  , 


DEVELOPMENTS  IN  SERIES. 


Ill 


then  the  coefficients  a  and  b  are  connected  by  the  relations 

K  -  log  a,; 

ctoK  =  «.; 

^aj)^  +  a  J),  =  2a,; 
3a,b,  +  2a,b,  +  a,&,  =  3a,; 
etc.  etc.  etc. 

1 


1 7.  Hence  show  that 


^  is  the  logarithm  of  the  sum  of 

1  —  X  ° 


an  infinite  series  whose  first  terms  are 

63.  Identity  of  Taylor's  and  Maclaurin's  I'heorems. 
These  two  theorems,  though  different  in  form,  are  identical 
in  principle. 

To  see  how  Taylor's  theorem  flows  from  Maclaurin's,  notice 
that  h  in  the  former  corresponds  to  x  in  the  latter.  The  de- 
rivatives with  respect  to  x  in  Taylor's  theorem  are  the  same 
as  the  derivatives  with  respect  to  7^,  and  if  we  suppose  A  =  0 
after  differentiation  Taylor's  form  of  development  can  be  de- 
rived at  once  from  Maclaurin's. 

Conversely,  Maclaurin's  theorem  may  be  regarded  as  a 
special  case  of  Taylor's  theorem,  in  which  we  take  zero  as  the 
original  value  of  the  variable,  and  thus  make  the  increment 
equal  to  the  variable.     That  is,  if  we  put/(u)  in  the  form 

/(O  +  ^'). 
and  then,  using  x  for  //,  develop  in  powers  of  x  by  Taylor's 
theorem,  wo  shall  have  Maclaurin's  theorem. 

64.  Cases  of  Failure  of  Taylor's  and  Maclaurin's 
Theorems.  In  order  that  a  development  in  powers  of  a  vari- 
ble  may  have  a  determinate  value  it  is  necessary  that  none  of 
the  coefficients  in  the  development  shall  become  infinite  and 
that  the  developed  series  shall  be  convergent. 

For  example,  cosec  x  cannot  be  developed  in  powers  of  x, 
because  when  x  =  0  the  cosecant  and  all  its  derivatives  be> 
come  infinite. 


:'f    ■■■ 


I 


>  1 1 

i 


U    i 


li 


B     I.' 


'     r' 


1       f 


112 


THE  DIFFERENTIAL  CALCULUS. 


65.  Extension  of  Taylor's  Theorem  to  Functions  of  Several 
Variables,     Let  us  have  the  function 

n=f{x,y).  (1) 

It  is  required  to  develop  this  function  when  x  and  y  both  re- 
ceive increments. 

Let  us  first  assign  to  x  the  increment  hy  and  suppose  y  to 
rema'n  constant.     We  then  have,  by  Taylor^s  theorem, 

/(^  +  7,,t,)  =  »  +  ^^+^,^-,  +  ^.3,  +  ...,{3) 

in  which  «,  y-,  etc.,  are  all  functions  of  y. 

Next,  assign  to  y  the  increment  k.  The  first  member  of 
(2)  will  become  /'(a;  -{-  h,  y  -^  k).  Developing  the  coefficients 
in  the  second  member  in  powers  of  k,  the  result  will  be: 

i*  will  be  c]ianged  into 


du 


-T-  E  i^a;?^  will  be  changed  into 


^       ,   d-Dji  k  ,   d\0^uk^   , 


d^u 


dy     1 


6/?/' 


T-j  E  jC^x'^*  will  be  changed  into 

^,     ,   d-^,'uk  ,  d'D^uk'   , 


etc. 


dy      1 
etc. 


dy""    /*; 
etc. 


Substituting  these  changed  values  of  the  coefficie-its  in  (2) 

it  will  become 

dii  k   .  d^u  ¥   .   d^u  k^ 


f{x  +  h,y  -^  k)  =  u  +  -^-  j-\-  - 


I         ,7^,3   Q  I        I        •     •     • 


ly  1   '   dy'-M   '   dy'dl 


du  h         d^u  h  k 


d'u   h  k' 
dxdyl  1   '   dxdy'l  2!~^'  *  * 
d'u    lek  .      dSi    h'k' 


^  dx'  21  ^  da-'dy'^l  1  ^'dx'dy'  2!  2! 

d'u  h' 
'^  dx'3r 


W  ''^- 


DEVELOPMENTS  IN  SERIES. 


113 


Thus  the  function  is  developed  in  powers  and  products  of 
the  increments  h  and  Tc. 

The  hiw  of  the  series  will  be  seen  most  clearly  by  using  the 
/)-notation.  For  each  pair  of  positive  and  integral  values  of 
m  and  n  we  shall  have  the  term 

"         m\  n\ 

If  we  collect  in  one  line  the  terms  of  the  development 
which  are  of  the  same  order  in  li  and  ky  we  shall  have; 


Order  of 
Terms. 


h 


h 


Ist.       D^U^  +  DyUzr, 


2d.    2>>^j  +  A^„?*^--j  +  i>»'w^j. 


''ill 


\  I 


I' 


fficieiits  in  (2) 


7i**  h^~^     h 

rth.  z>j-w-j  +  ^*''~'^w^«77:_-iyi  x  +  •  •  • 


EXERCISES. 


I.  Show  that  in  the  preceding  development  the  terms  of 
the  rth  order  may  be  written  in  the  form 

=-j,  (^  j,  etc.,  denoting  the  binomial  coefficients  as  in  §  5. 


2.  Extend  the  development  to  the  case  of  three  independent 
I  variables,  and  show  that  the  terms  to  the  second  order  in- 
iclusive  will  be  as  follows ; 


--S 


ill 


hi   i 


III 


:  i 

i  I 


V    i? 


114 


If 


THE  DIFFERENTIAL  CALCULUS. 


u=f{x,y,  z), 
then  /•  (x  -{-  h,  y  -{-  k,  z  +  I)  —  u 

+  D^uD^u '  hi  4-  DyiiD^u '  hi, 

66.  Hyperbolic  Functions.  The  sine  and  cosine  of  an 
imaginary  arc  may  be  found  as  follows:  In  the  developments 
for  sin  x  and  cos  x,  namely, 

3"!  +  5! 


sin  2;  =:.  a;  -  I,  +  1^  -  ,  . 


1        x"    ^    X* 
cos  re  =  1  -  -J  +  -J  -  ...  , 


let  us  put  yi  for  a:,     (i  e  4/—  1).     We  thus  have 
8m?/i  =  i'^;?/  +  ^j  +  ^j+  .  .  . 

cos  y  t  =  1  +  |j  4-  |j  4- .  ... 


V 


(1) 


We  conclude: 

The  coaine  of  a  purely  imaginary  arc  is  real  and  greater 
than  U7iity,  while  its  sine  is  jmrely  imaginary. 
We  find  from  (1), 

cos  yi  4-  *  sin  yi  =  1  —  y  -\-  |-  —  etc.  =  e~^; 

?/» 

cos  yi  —  i  sin  yi  =  1  -\-  y  -\-  -j-  4-  etc.  =  e"; 

z  I 

and,  by  addition  and  subtraction, 

cos  yi  =  ^{e~'"  -\-  e^)'j 

i  sin  yi  =.  ^[g~^  —  e"); 

sin  yi  —  \i{e^  —  e~^). 

The  cosine  of  yi  is  called  the  hyperbolic  cosine  of  y,    | 
and  is  written  cosh  y,  the  letter  h  meaning  ^Miyperbolic' 


jy 


DEVELOPMENTS  IN  SERIES. 


115 


^D^DyU-hh 


cosine  of   an 
developments 


(1) 


.al  and  greater 


=  6-"; 


=  6"; 


c  cosine  of  y, 

hyperbolic." 


The  real  factor  in  the  sine  of  yi  is  called  the  hyperbolic 
sine  of  y,  and  \l  written  sinh  y. 

Thus  the  liyperbolic  sine  and  cosine  of  a  real  quantity  are 
real  functions  defined  by  the  equations 

sinh  y  =  ^a"  —  c~^); 
cnshy  =  ^{e"  -{-e-") 

By  analogy,  we  introduce  tlie  additional  function 


-'h) 
'-").) 


(1) 


(,V  (,-V 

tanh  y  —  — — -. 

The  differentiation  of  these  expressions  gives 

d  sinh  y  ,  d  cosh  y        .   , 

i — ■-  =  cosh  y:    7-  —  =  smh  y: 

dy  -^^         dy  ^ 


(2) 


d  tanh  y  =  — ~ 
•^       cosh 


y 


They  also  give  the  relations 

cosh'  y  —  sinh'  y  =  1.  (3) 

fn verse  Hyperbolic  Functions,     When  we  form  the  inverse 
function,  we  may  put 

u  E  cosh  y. 

Then,  solving  the  equation 

e^  +  e-"  =  3  cosh  y  =  2w, 

we  find  6"  =  2^  ±  Vu"^  —  1. 

Hence 


y  =  log  {n  ±  Vu""  —  1)  =  cosh  <-^> 

In  the  same  way,  if  we  put 

u  ~  sinh  y, 
we  find 


u. 


(4) 


y  —  log  {u  ±  Vii"  -f  1)  =  sinh<-*)  u. 


(5) 


From  the  equations  (2)  and  (3)  we  find,  for  the  derivatives 
of  the  inverse  functions: 


ft 

^  ill 

i 

'11 

^ ;  1 1 


P  ' 


m 

i-\i 


*M 


m 


I  -n' 


lit 


116 

TUF.  DIFFEUENTIAL  GALGUL  US. 

When 

y  =  cosh^""  ^>  Uf     or    u  =  cosh  y, 

ii 

then 

^iy  _.     1 

du        i^tt^  _  1 

, 

When 

y  =  sinh^~  ^^  w,     or    u  =  sinh  y. 

then 

dy  _       1 

du        Vu*  4- 1 

frii 


.1L 


(6) 


(7) 

Kemark.  The  above  functions  are  called  hyperbolic  be- 
cause sinh  y  and  cosh  y  may  be  represented  by  the  co-ordinates 
of  points  on  an  equilateral  hyperbola  whose  semi-axis  is  unity. 
The  equation  of  such  an  hyperbola  is 

x^-f  =  1, 
which  is  of  the  same  form  as  (3). 

EXERCISES. 

I.  By  continuing  the  differentiation  begun  in  (3)  prove 
the  following  equations: 

Bx  sinh  X  =  sinh  x; 

D^  cosh  X  =  cosh  x\ 

D^^~^'  sinh  x  —  sinh  x. 


etc. 


etc. 


2.  Develoj)  sinh  .r,  as  defined  in  (1),  in  powers  of  x  by  Mac- 
laurin^s  theorem. 

Ans.  sinh  ^'  =  2/  +  of  +  «"?+•••  • 

3.  Develop    sinh    {x  -\-  h)   and   cosh  {x  -f-  //-)   by  Taylor^s 
theorem  and  deduce 

sinh  {x-\-h)  =  sinh  xll  -\-  -^  -{-  .  .  .1+  cosh  xlx  +  -^  +  .  .  .  J 

=  sinh  X  cosh  h  -\-  cosh  x  sinh  //; 
cosh  (x-\-h)  —  cosh  x  cosh  h  -j-  sinh  x  sinh  h. 


I 


MAXIMA    AND   MINIMA, 


117 


(6) 


yperboUc  be- 
3  co-ordinates 
-axis  is  unity. 


in  (3)  prove 


•s  of  X  by  Mac- 
)   by  Taylor's 


CHAPTER  IX. 

MAXIMA  AND   MINIMA  OF   FUNCTIONS  OF  A 
SINGLE  VARIABLE. 

67.  Def.  A  maximum  value  of  a  function  is  one  which 
is  greater  than  the  values  immediately  preceding  and  follow- 
ing it. 

A  minimum  value  is  one  which  is  less  than  the  values 
immediately  preceding  and  following  it. 

Remark.  Since  a  maximum  or  minimum  value  does  not 
mean  the  greatest  or  least  possible  value,  a  function  may 
have  several  maxima  or  minima. 

68.  Problem.    Having  given  a  function 

y  =  <P{^)y 

it  is  required  to  find  those  vahtes  of  x  for  which  y  is  a  maxi- 
mum or  a  minimum. 

Let  us  assign  to  x  the  increments  +  h  and  —  //-,  and  develop 
in  powers  of  h.     We  shall  then  have 

."..(.  +  .)  =.  +  |^  +  gft  +  otc. 

In  order  that  the  value  of  y  =  (f)(x)  may  be  a  minimum,  it 
must,  however  small  we  suppose  h,  be  less  than  either  y'  or 
y".    That  is,  the  expressions 

,  dy  h    ,   d'y  7^'         , 


'( 


1  : 


Hi 


'  'A 


118 


THE  DIFFERENTIAL  CALCULUS. 


11 


I 


li 


iltl 
must  both  bo  positive  as  h  approaches  zero.     But  if  —-  is 

finite,  h  may  always  be  made  so  small  that  the  terms  in  /*' 
shall  bo  less  in  absolute  magnitude  than  those  in  h  (§  14),  and 
the  condition  of  a  minimum  cannot  bo  satisfied.  We  must 
therefore  have,  as  the  first  condition, 


% = '^'(^J  =  0- 


(1) 


By  solving  this  equation  with  respect  to  x  will  be  found  a 
value  of  X  called  a  critical  value. 

The  same  reasoning  applies  to  the  case  of  a  maximum,  so 
that  the  condition  (1)  is  necessary  to  either  a  maximum  or  a 
minimum.     Supposing  it  fulfilled,  we  have 


y  -y 


dx'1-2       drU-a-3 


+  etc.; 


y  -y- 7/? r"2  + 1? r~2" 3 "^ ^*^- 

Since  U^  is  positive,  the  algebraic  sign  of  these  quantities. 


as  1)  approaches  zero,  will  be  the  same  as  that  of 


(Vy 
dx'' 


When  this  second  derivative  is  positive  for  the  critical  value 
of  X,  y,  being  less  than  y'  or  y",  will  be  a  minimum. 

When  negative,  y  will  be  greater  than  either  y'  or  y",  and 
so  will  be  a  maximum. 

We  therefore  conclude: 

dv  d^v 

Conditions  of  minimum:     ~-  =  0;     -jtu  positive. 


dx 


Conditions  of  maximum: 


%=^'  w-'"^"^"'- 


We  have,  therefore,  the  rule: 

.Equate  the  first  derivative  of  the  function  to  zero.  This 
equation  will  give  one  or  more  values  of  the  independent  vari- 
able, called  critical  vahies,  and  thence  corresjjonding  values  of 
the  function. 


MAXIMA  AND  MINIMA. 


110 


Suhstitute  the  criiical  values  in  the  exprcHnion  for  the  second 
derivative.  When  the  result  is  positive,  the  function  is  a 
minimvm;  when  negative,  a  maximum. 

Exceptional  Cases.  It  muy  happen  that  the  second  deriva- 
tive is  zero  for  a  critical  vaUio  of  x.     Wo  shall  then  have 


y  -y 


y"  -  y 


dx'  31'^  dx'4l      ^^^-^ 

dy  h^  ,(]*yh' 

dx'dl'^  dx*  4!  +  ^^^-' 


'     f. 


and  there  can  be  neither  a  maximum  nor  a  minimum  unless 

-7-^  =  0.     If  this  condition  is  fulfdled,  y  will  be  a  maximum 

dx'  '' 

■when  the  fourth  derivative  is  negative;  a  minimum  when  it 
is  positive. 

Continuing  the  reasoning,  we  are  led  to  the  following  ex- 
tension of  the  rule: 

Find  the  first  derivative  in  order  which  does  not  vanisJi 
for  a  critical  value  of  the  indej^endent  variable.  If  this  de- 
rivative is  of  an  odd  order,  there  is  neither  a  viaximwn  nor  a 
minimum;  if  of  an  even  order,  there  is  a  minimnm  when  the 
derivative  is  positive,  a  maximum  when  it  is  negative. 

The  above  reasoning  may  bo  illustrated  by  the  graphic  rop- 
resontation  of  the  function.  When  the  ordinate  of  the  curve 
is  u  maximum  or  a  minimum  the  tan.4ont  will  bo  iiarallcl  to  the 
axis  of  abscissas,  and  the  anglo  which  it  makes  with  this  axis 
will  change  from  positive 
to  negative  at  a  point  hav- 
ing a  maximum  ordinate, 
and  from  negative  to  posi- 
tive at  a  point  having  a 
minimum  ordinate. 

For  example,  in  the  fig-  ^'°-  ^^• 

urc  a  minimum  ordinate  occurs  at  the  point   Q,  and  maxi- 
mum ordinates  at  /*  and  R. 


■  t 


;  I 


,    V  1 


m 


M' 


! 


ll 


.  t 

I 

i 

120  THffJ  niFFKHENTIAL  CALCULUS. 


EXAMPLES  AND  EXERCISES. 

I.  Find  the  maximum  and  minimnm  values  of  the  exprtra- 
sion 

p  =  2x'  +  3x'  -  3Gx  +  15. 

By  differentiation, 

^L  =  G.1-'  +  Gx  -  30; 
dx 

^  =  13:r  +  6. 
ax 

Equating  the  first  derivative  to  zero,  we  have  the  quadratic 
equation 

x'  -{-  X  —  G  =  0, 

of  which  the  roots  are  x  =  2  and  x  =  —  3, 

(Vx 
The  vahies  of  -p,  are  +  30  and  —  30. 

Hence  x  =  2  gives  a  minimum  value  of  y  =  —  29; 

X  =  —  3  gives  ii  maximum  vahie  oi  y  =  -{-  95. 

Fi.  id  the  maximum  and  minimum  vaUies  of  the  following 
funciions: 


2.  x'  4-  3x'  -  2ix  +  9. 

X 


4'  y  = 


l+x 


»• 


3,  x'  —  3x  4-  5. 
_x'^  —  X  -j-  1 

5-  y  -  ^r+~x'^^' 


]osx 
6.  y  =  -^~. 
^  X 


7.  y 


_  log^ 


/vTl 


9.  y  =  sin  2x  —  X. 

II.  y  —  [x  —  rt')(,r  —  Z>)', 

(a:  —  a){x  —  h) 
13.  y  =  7 r) (. 

15.    ?/  =  cos  ?i.T. 

17.  y  ~  sin  ?ia;. 
^ ?w.  A  maximum  when  x  =  -f-cos  x, 
1  -\-  X  tan  a;'  A  minimum  when  x  ~  —cos  x. 


8.  y 
10.  y:^{x  +  l){x-2)\ 

_  {x  +  3)' 
'^*  •^^~  (.'?;+ 2)'* 
14.  y  =  cos  2.^*. 
16.  7/  -  sin  3x. 

'8.  y  = 


h 


MAXIMA  AND  MIA' I  MA. 


121 


19.  ^  =  sm  X  cos  X, 

sin  X 

2i»  y  =  z — r~i • 

^      1  +  tun  X 


20.  y  =  sin'  X  cos  x, 

cos  2; 
32.  y  = 


1  -f-  tan  x' 

23.  The  sura  of  two  adjacent  sides  of  a  rectangle  is  equal 
to  a  fixed  line  a.  Into  what  parts  must  a  be  divided  that  the 
rectangle  may  bo  a  maximum?  A718.  Each  part  =  ia. 

Note  that  the  expression  for  the  area  i«  x{a  —  x). 

24.  Into  what  parts  must  a  number  be  divided  in  order 
that  the  product  of  one  part  by  the  square  of  the  other  may 
be  a  maximum?  A71S.  Into  jiarts  wliose  ratio  is  1  :  3. 

Note  that  if  a  be  the  number,  the  parts  may  be  called  x  and  a  —  x. 

25.  Into  what  two  parts  must  a  number  be  divided  in  order 
that  the  product  of  the  miXi  power  of  one  i)art  into  the  nth 
power  of  the  other  may  be  a  maximum? 

Ans.  Into  parts  whose  ratio  is  m  :  n, 

26.  Show  that  the  quadratic  function  ax^  -\-J)x-\-c  can  have 
but  one  critical  value,  and  that  it 
will  depend  upon  the  sign  of  the 
coefficient  a  whether  that  value 
is  a  maximum  or  a  minimum. 

27.  A  line  is  required  to  pass 
through  a  fixed  point  P,  whose 
co-ordinates  are  a  and  h  in  the 
plane  of  a  pair  of  rectangular 
axes  OX  and  OY.  "What  angle 
must  the  line  make  with  the  axis 
of  X,  that  the  area  of  the  triangle  XYO  maybe  a  minimum? 
Show  also  that  P  must  bisect  the  segment  XY, 

Express  the  intercepts  which  the  line  cuts  off  from  the  axes  in  terms  of 
a,  b  and  the  variable  angle  a.  The  half  product  of  these  intercepts  will 
be  the  area. 

We  shall  thus  find 


FlO.  13. 


«'i 


,\  'I 


;  n 


I 


2  Area  =  {a-\-b  cot  cc)ip  -f-  a  tan  a)  =  2ad  -{-  a^  tan  a  + 


tan  a 


a 

f  ' 

1  T" 

1 

I 

5 

1 1.  ■ 

;!:' 

1! 

h 

I' 

L 

iili; 

i 

1 

i  ^ 

1 

r 

i 

'in, 

v:' 

f 

l> 


I  Jli 


I  ■'. 


122 


7!ffiS^  DIFFERENTIAL  CALCULUS. 


Then,  taking  tan  a:  —  <  as  the  independent  variable,  we  readily  find,  for 
the  critical  values  of  t  and  or. 


«=  ± 


a' 


or    a  sin  a  =  ±  6  cos  a. 


It  is  then  to  be  shown  that  both  values  of  t  give  minima  values  of  the 
area  ;  that  the  one  minimum  area  is  2ad,  and  the  other  zero  ;  that  in  the 
first  case  the  line  YX  is  bisected  at  P,  and  in  the  other  case  passes 
through  0. 

28.  Show  by  the  preceding  figure  that  whatever  be  the  an- 
gle XO  Y,  the  area  of  the  triangle  will  be  a  minimum  when 
the  line  turning  on  P  is  bisected  at  P. 

The  student  should  do  this  by  drawing  through  P  a  line  making  a 
small  angle  with  XPT.  The  increment  of  the  area  XOY  will  then  be 
the  difference  of  the  two  small  triangles  thus  formed.  Then  let  the  small 
angle  become  infinitesimal,  and  show  that  the  increment  of  the  area 
XOFcan  become  an  infinitesimal  of  the  second  order  only  when  PX=z 
PY. 

29.  A  carpenter  has  boards  enough  for  a  fence  40  feet  in 
length,  which  is  to  form  three  sides  of  an  enclosure  bounded 
on  the  fourth  by  a  wall  already  built.  What  are  the  sides 
and  area  of  the  largest  enclosure  he  can  build  out  of  his  ma- 
terial? Ans.  10  X  20  feet  =  200  square  feet. 

30.  A  square  piece  of  tin  is  to  have  a  square  cut  out  from 
each  corner,  and  tlio  four  projecting  flaps  are  to  be  bent  up  so 
as  to  form  a  vessel.  What  must  be  the  side  of  the  part  cut 
out  that  the  contents  of  the  vessel  may  be  a  maximum? 

Ans.  One  sixth  the  side  of  the  square. 

31.  If,  in  this  case,  the  tin  is  a  rectangle  whose  sides  are 
2a  and  2h,  show  that  the  side  of  the  flap  is 


32.  What  is  the  form  of  the  rectan- 
gle of  greatest  area  which  can  be  drawn 
in  a  semicircle? 

Note  that  if  r  be  the  radius  of  the  circle, 


and  X  the  altitude  of  the  rectangle,  |/r*  —  «* 
will  be  half  the  base  of  the  rectangle. 


Fio.  14. 


MAXIMA  AND  MINIMA, 


123 


lily  find,  for 


values  of  the 

;  that  in  the 

case  passes 

be  the  an- 
muin  when 

ine  making  a 
"  will  then  he 
1  let  the  small 
t  of  the  area 
y  when  PX  = 

e  40  feet  in 
ire  bounded 
re  the  sides 
t  of  his  ma- 
juare  feet, 
ut  out  from 
DO  bent  up  so 
the  part  cut 
imum? 
he  square, 
ose  sides  are 


-^ 

1 

X                             '9^B 

/ 

\                  ''^H 

\                         '-^^H 

y 

\               '  -'-'i^^H 

f 

\               ^^^H 

/ 

\                   l^M. 

/ 

\               '^^B 

f 

\           '  '^^B 

/ 

o 

\           'Wk 

69.  Case  when  the  function  which  is  to  he  a  maximum  or 
minimum  is  expressed  as  a  function  of  two  or  more  variables 
connected  hy  equations  of  condition. 

The  function  which  is  to  be  a  maximum  or  minimum  may 
be  expressed  as  a  function  of  two  variables,  x  and  y,  thus: 

u  =  <f)(x,  y),  (1) 

If  x  and  y  are  independent  of  each  other,  the  problem  is 
different  from  that  now  treated. 
If  between  them  there  exists  some  relation 

A^>  y)  =  0,  (3) 

we  may,  by  solving  this  equation,  express  one  in  terms  of  the 
other,  say  y  in  terms  of  x.  Then  substituting  this  value  of 
y  in  (1),  to  will  be  a  function  of  x  alone,  which  we  may  treat 
as  before. 

It  may  be,  however,  that  the  solution  of  the  equation  (2) 
will  be  long  or  troublesome.  We  may  then  avoid  it  by  the 
method  of  §  41.    From  (1)  we  have 

dii  _  ldu\       fdu\dy 
dx  ~~  \dx  I       \dy  Idx ' 

and  from  (2)  we  have,  by  the  method  of  §  37, 

dy  ^      BJ^ 

dx  Byf 

Substituting  this  value  in  the  preceding  equation,  we  shall 

have  the  value  of  -7-,  which  is  to  be  equated  to  zero.     The 

equation  thus  formed,  combined  with  (2),  will  give  the  critical 
values  of  both  x  and  y,  and  hence  the  maximum  or  minimum 
value  of  w. 


m 


\u: 


;'!'' 


io.  U. 


124 


THE  DIFFERENTIAL  CALCULUS. 


I 


EXAMPLES    AND    EXERCISES. 

1.  To  find  the  form  of  that  cylinder  which  has  the  maxi- 
mum volume  with  a  given  extent  of  surface. 

The  total  eirtent  of  surface  includes  the  two  ends  and  the  convex 
cylindrical  surface.  If  ?•  be  the  radius  of  the  base,  and  h  the  altitude, 
we  shall  have : 

Area  of  base,  nr"^. 

Area  of  convex  surface,  ^icrh. 

Hence  total  surface  =  27r(r'  +  ^^)  =  const.  =  a.  (a) 

Also,  volume  =  itr^h.  (6) 

Putting  u  for  the  volume,  we  have,  from  (&), 

du      f.     ,   .       jdh  ^ 

dr  dr 

From  (a)  we  find 

"Whence 

Equating  this  to  zero,  we  find  that  the  altitude  of  the  cylinder  must  be 
equal  to  the  diameter  of  its  base. 

2.  Find  the  shape  of  the  largest  cylindrical  tin  mug  which 
can  be  made  with  a  given  weight  of  tin. 

This  problem  differs  from  the  preceding  one  in  that  the  top  is  sup- 
posed to  be  open,  so  that  the  total  surface  is  that  of  the  base  and  con- 
vex portion. 

Ans.  Altitude  =  radius  of  bottom. 

3.  Find  the  maximum  rectangle  which  can  be  inscribed  in 
a  given  ellipse. 

If  the  equation  of  the  ellipse  is  JV  -j-  aV  =  »'^*.  the  sides  of  the 
rectangle  are  2x  and  2y.  Hence  the  function  to  be  a  maximum  is  4ry, 
subject  to  the  condition  expressed  by  the  equation  of  the  ellipse.  This 
condition  gives 

dy  h^x 


dh 

n 

+  2r 

dr~ 

• 

r     ' 

du 
dr~ 

nrh 

-  2nr\ 

dx 


a?y 


MAXIMA  AND  MINIMA. 


126 


We  shall  find  the  rectangle  to  be  a  maximum  when  its  sides  are 
proportional  to  the  corresponding  axes  of  the  ellipse;  each  side  is  then 
equal  to  the  corresponding  axis  divided  by  V^. 

4.  Find  the  maximum 
rectangle  which  can  be 
inscribed  in  the  segment 
of  a  parabola  whose  semi- 
parameter  is  p,  cut  off  by 
a  double  ordinate  whose 
distance,  OX,  from  the 
vertex  is  a.  Show  also 
that  the  ratio  of  its  area 
to  that  of  the  circum- 
scribed rectangle  is  con- 
stant and  equal  to 

2:i/27. 

By  taking  x  and  y  as  in  the  pi<j.  15. 

figure,  a  —  X  will  be  the  base 

of  the  rectangle,  and  we  shall  have  2y  for  its  altitude.  Hence  its  area 
will  be  3y(a  —  x),  while  x  and  y  will  be  connected  by  the  equation  of  the 
parabola,  y^  =  2px. 

5.  Find  the  cone  of  maximum  volume  which  shall  have  a 
given  extent  of  conical  surface. 

Ans.  Alt.  =  radius  of  base  X  ^. 

6.  Find  the  volume  of  the  maximum  cylinder  which  can  be 
inscribed  in  a  given  right  cone,  and  show  that  the  ratio  of  its 
volume  to  that  of  the  cone  is  4  :  9. 

7.  Find  the  cylinder  of  maximum  cylindrical  surface  which 
can  bo  inscribed  in  a  right  cone. 

Ans.  Alt.  of  cylinder  =  |  alt.  of  cone. 

8.  Find  the  maximum  cone  which  can  be  inscribed  in  a 
given  sphore. 

If  wc  make  a  central  section  of  the  sphere  through  the  vertex  of  the 
cone,  the  base  and  slant  height  of  the  cone  will  be  the  base  and  equal 


.i 


I 


m 


m 


in 


126 


THE  DIFFERENTIAL  CALCULUS. 


s 


Hi ,  ■;! 

im 


I  i! 


sides  of  an  isosceles  triangle  inscribed  in  the  circular  section.    Thus  the 
equation  between  the  base  and  altitude  of  the  cone  can  be  obtained. 

Ans.  Alt.  =  ^  radius  of  sphere. 

9.  Find  the  maximum  cylinder  which  can  be  inscribed  in 

an  ellipsoid  of  revolution. 

2 
Ans.  Alt.  =  — -  of  axis  of  revolution. 

V3 

10.  Find  the  cone  of  maximum  conical  surface  which  can 
be  inscribed  in  a  given  sphere. 

11.  Of  all  cones  having  the  same  slant  height,  which  has 
the  maximum  volume? 

12.  A  boatman  3  miles  from  the  shore  wishes,  by  rowing 
to  the  shore  and  then  walking,  to  reach  in  the  shortest  time 
a  point  on  the  beach  5  miles  from  the  nearest  point  of  the 
shore.  If  he  can  pull  4  miles  an  hour  and  walk  5  miles  an 
hour^  to  what  point  of  the  beach  should  he  direct  his  course? 

Ans,  4  miles  from  the  nearest  point  of  the  shore. 

Express  the  whole  time  required  in  terms  of  the  distance  x  of  his  point 
of  landing  from  the  nearest  point  of  the  shore. 

13.  Find  the  maximum  cone  which  can  be  inscribed  in  a 
paraboloid  of  revolution,  the  vertex  of  the  cone  being  at  the 
centre  of  the  base  of  the  paraboloid. 

Ans.  Alt.  =  I  alt.  of  paraboloid. 

14.  Find  the  maximum  cylinder  which  can  be  described  in 
a  paraboloid  of  revolution. 

15.  Find  the  rectangle  of  maximum  perimeter  which  can 
be  inscribed  in  an  ellipse. 

16.  On  the  axis  of  the  parabola  y^  =  2px  a  point  is  taken 
at  distance  a  from  the  vertex.  Find  the  abscissa  of  the  near- 
est point  of  the  curve. 

Begin  by  expressing  the  square  of  the  distance  from  the  fixed  point  to 
the  variable  point  (x,  y)  on  the  parabola. 

17.  Determine  the  cone  of  minimum  ,  jlume  which  can  be 
circumscribed  around  a  given  sphere. 


MAXIMA  AND  MINIMA. 


127 


iS.  Determine  the  cone  of  minimum  conical  surface  which 
can  be  circumscribed  around  a  given  ophere. 

19.  Find  that  point  on  the  line  joining  the  centios  of  two 
circles  from  which  the  greatest  length  of  the  combined  cir- 
cumferences will  be  visible. 

20.  Find  that  point  on  the  line  joining  the  centres  of  two 
spheres  of  radii  a  and  h  respectively  from  which  the  greatest 
extent  of  spherical  surface  will  be  visible. 

Alls,  The  point  dividing  the  central  line  in  the  ratio  a  :  h  . 

21.  Show  that  of  all  circular  sectors  described  with  a  given 
perimeter,  that  of  maximum  area  has  the  arc  equal  to  double 
the  radius. 

22.  A  ship  steaming  nortb  12  knots  an  hour  sights  an- 
other ship  10  miles  ahead,  steaming  east  9  knots.  What  will 
be  the  least  distance  between  the  ships  if  each  keeps  on  her 
course,  and  at  what  time  will  it  occur? 

Ans.  Time,  32  min. ;  distance,  6  miles. 

23.  What  sector  must  be  taken  from  a  p'iven  circle  that  it 
may  form  the  curved  surface  of  a  cone  of  maximum  volume? 

Ans.    V^  of  the  circle. 

24.  A  Norman  window,  consisting  of  a  rectangle  sur- 
mounted by  a  semicircle,  is  to  admit  the  maximum  amount 
of  light  with  a  given  perimeter.  Show  that  the  base  of  the 
rectangle  must  be  double  its  altitude. 


i  ^  Ml 


N;li" 


if 

,ii  fl 

Ml      i 


128 


THE  DIFFERENTIAL  CALCULUS. 


i 


!    II 


I 


1    «! 


W: 


CHAPTER  X. 

INDETERMINATE    FORMS. 

70.  Let  us  consider  the  fraction 

a:' -9 


0W  = 


a; -3' 


(1) 


For  any  value  we  may  assign  to  x  there  will  be  a  definite 
value  of  ({){x)  found  by  dividing  the  numerator  of  the  frac- 
tion by  the  denominator. 

To  this  statement  there  is  one  exception,  the  case  ot  a:  =  3. 
Assigning  this  value  to  x,  we  have 

0(3)  =  f 

Now,  the  quotient  of  two  zeros  is  essentially  indeterminate. 
For  the  quotient  of  any  two  quantities  is  that  quantity 
which,  multiplied  by  the  divisor,  will  produce  the  dividend. 
But  any  quantity  whetever  when  multiplied  by  0  will  pro- 
duce 0.  Hence,  when  divisor  and  dividend  are  both  zero, 
any  quantity  whatever  may  be  their  quotient. 

But  when  we  consider  the  terms  of  the  fraction,  not  as  ab- 
solute zeros,  but  as  quantities  approaching  zero  as  a  limit, 
then  their  quotient  may  approach  a  definite  limit.  We  then 
regard  this  limit  as  the  value  of  the  fraction  corresponding 
to  zero  values  of  its  terms. 

As  another  example,  consider  the  quantity 


X 


2 


X 


4* 


We  may  compute  the  value  of  this  expression  for  any  value 
of  X  except  2.  When  x  =  2  the  ucrms  will  both  become  in- 
finite.    Siuce  if  any  quantity  whatever  be  added  to  an  infinite 


INDETERMINATE  FORMS. 


129 


ot  ic  =  3. 


terminate. 
b  quantity 
dividend. 
I  will  pro- 
both  zero, 

not  as  ab- 
s  a  limit. 
We  then 
•esponding 


any  value 
lecome  in- 
an  infinite 


va 


::l 


the  sum  will  be  infinite,  it  follows  that  any  quantity  what- 
ever may  bo  the  difference  of  two  infinites. 

There  are  several  other  indeterminate  forms.  The  follow- 
ing are  the  principal  ones  which  take  an  algebraic  form: 

-;    — ;     OXoo;     oo  -  oo ;     0";     oo";     l". 

•71.  Evahiation  of  the  Form  %.  In  many  cases  the  inde- 
terminate character  of  an  expression  may  be  removed  by 
algebraic  transformation.  For  example,  dividing  both  terms 
of  the  fraction  (1)  by  x  —  3,  it  becomes  a;  -[-  3,  a  determinate 
quantity  even  for  ic  =  3.     Again,  the  expression  (2)  can  be 

reduced  to  the  form  — -— r,  which  becomes  i  when  a;  =  2. 

a;+  2 

The  general  method  of  dealing  with  the  first  form  is  as 
follows:  Let  the  given  fraction  be 

^{xY 

and  let  it  be  supposed  that  both  terms  of  this  fraction  vanish 
when  X  —  a,  s,o  that  we  have 

0(rt)  =  0    and    tp{a)  =  0.  (3) 

Put  7i=x  —  a,  and  develop  the  terms  in  powers  of  h  by 
Taylor's  theorem.     We  shall  then  have 

^{x)  =  <p(a  +  A)  =  0(«)  +  h<p\a)  +  3^0"(«)  +  .  .  .  ; 

tp{x)  =  iia  +  h)  =  tp(a)  +  hf(a)  +  ^^"(«)  +  .  .  •  J 
whence,  for  the  value  of  the  fraction  (comp.  Eq.  (3)), 

^  ^  0'w  + 1^0"(«)  +  •  ■  ■  ^^^ 

Now,  when  7i  approaches  zero  as  a  limit,  the  value  of  this 
fraction  approaches 

0>) 


■  f  i: 


11 


1 1 

ill 

:  !  !  1 


t 


';  fi 


)i.:: 


'  i 


'■I 

ii 


si, 


i! 


s 

ii  i 

lIlN    : 


130 


THE  DIFFERENTIAL  CALCULUS, 


as  a  limit,  which  is  therefore  the  required  limit  of  the  frac- 
tion when  both  its  members  approach  the  limit  zero. 

It  may  happen  that  0'(«)  and  ip'(a)  both  vanish.  In  this 
case  the  required  limit  of  the  fraction  in  (4)  is  seen  to  be 

In  general:  The  required  limit  is  the  ratio  of  the  first  pair 
of  derivatives  of  like  order  tohich  do  7iot  both  vanish. 

If  the  first  derivative  which  vanishes  is  not  of  the  same 
order  in  the  two  terms, — for  example,  if,  of  the  two  quantities 
0'(«)  and  ^'(«),  one  vanishes  and  the  other  does  not, — then 
the  limit  of  the  fraction  will  be  zero  or  infinity  according  as  the 
vanishing  derivative  is  that  of  the  numerator  or  denominator. 

Remark.  It  often  happens  that  the  terms  of  the  fraction 
can  be  developed  in  the  form  (4)  without  forming  the  succes- 
sive derivatives.  It  will  then  be  simpler  to  use  this  develop- 
ment instead  of  forming  the  derivatives. 


I. 


X  —  a 


EXAMPLES    AND    EXERCISES, 
for     X  =  «.* 


0(a;)  =  a;'  —  «';     (t>'(x)  =2x;    .  • .  0'(a)  =  2a; 
ip{x)  =  X  —  a;     ^'(x)  =  1;      .•,  ^'(«)  =  1. 


lim. 


x  —  a 


(x  =  a)  —  2«, 


a  result  readily  obtained  by  reducing  the  fraction  to  its  lowest 
terms. 


2.  — ^-rr    for    x=l. 


x-1 


,  —  m 


X 


for    x  =  0. 


Ans.  1. 
Ans.  2. 


*  Using  strictly  the  notation  of  limits,  we  should  define  the  quantity 
sought  as  the  limit  of  the  fraction  when  x  approaches  ihe  limit  a.  But 
no  confusion  need  arise  from  regarding  the  limit  of  the  fraction  as  its 
value  for  a;  =  a,  as  is  customary. 


INDETERMINATE  FORMS. 


a;  —  sm  a     ,        ,  . 

4.  j for    {x  =  0).  -4m«.  |. 


131 


Here  the  successive  derivatives  of  the  terms  are: 

<})'(x)  =  1  —  cos  x)    <p"{x)  =  sin  x;    0'"(a;)  =  cos  x, 
tb\x)  =  3a;';  f'\x)  =  6x;        tp"\x)  =  6. 

The  third  derivatives  are  the  first  ones  which  do  not  vanish 
for  x  =  0. 

a"  —  ¥ 


X 


for    a;  =  0. 


Ans,  logrt— logJ=log^. 


.    tan  a:  —  sin  a;    . 

6.  r for    a;  =  0.       ^«s.  3. 


7 
8. 


a;  —  sm  a; 


1  —  cos  nx 
a"  —  a 


for    a;  =  0. 


a;-l 
o*  —  J* 


9.  ^ 

a;  —  1 

sin  a;  —  sin  « 


10. 


II. 


X  —  a 


for    a;  =  1. 
for    a;  =  1. 

for    x  =  a. 


A  2 

Ans,  — ,. 

-4%s.  a  log  «, 

-4«s.  a  log  a  —  J  log  h, 

Ans,  cos  a. 


tan 


y  —  tan «     ,  .         sec' « 

^^^^ s—    for    if  =  a,      A71S.  ^r—. — . 


cos  y  —  cos  a 


2  sin  a 


^^    log  (1  +  .)  +  log  (1  -  .)     ,„^     ^  ^  ^      ^^^_        ^_ 
COS  a;  —  sec  a;  ' 

log («  +  ^) - iog_(«-^  j^^  ^^j,    ^^_a 

a;  a 

sin  2a;  +  2  sin'  a;  —  2  sin  a;   ,  .         . 

14.  ■ 5 for    X  =  0.      Ans.  4. 

cos  a;  —  cos  x 

e'  —  e-"  —  2x 


15. 


a;  —  sm  a; 


for    a;  =  0.     -4ws.  1. 


16.     ,      .-—f — r—    for    y  =  0.      ^Ms.  2. 
log  (1  -f  y) 

1  —  sin  a;  —  cos  a;  -f  log  (i  +  a;)  ,      ,        ..        . 

17. — :r  for  (x  =  0).    Ans.  0. 

e*  —  1  —  a;  ^  ' 


i 


I 


I! 

'Hi 


i     i 


m  !! 


i,  1'! . 

1^  iii 


1 
iiil 


li      I' 


lir 


132 


TE^  DIFFERENTIAL  CALCULUS. 


00 


78.  Forms  —  rt^t^  0  X  oo.    Those  forms  may  be  reduced 

to  the  preceding  one  by  a  simple  transformation.     Any  frac- 

JV  1  —  D 

tion  -7--  may  be  written  in  the  form  ~ — ^>.    If  N  and  D  both 

become  infinite,  1-4-7)  and  1  ~r  N  will  both  become  infini- 
tesimal, and  thus  the  indeterminate  form  of  the  fraction  will 

bet. 

Again,  if  of  two  factors  A  and  B,  A  becomes  infinitesimal 
while  B  becomes  infinite,  we  write  the  product  in  the  form 

; ^,  and  then  it  is  a  fraction  of  the  first  form. 

I  -i-  B 

But  this  transformation  cannot  always  be  successfully  ap- 
plied unless  the  term  which  becomes  infinite  does  so  through 
having  a  denominator  which  vanishes.  For  example,  let  it 
be  required  to  find  the  limit  of 

a;'"(log  xY 

for  X  ^0.  Here  x^  approaches  zero,  while  log  x,  and  there- 
fore (log  xY,  becomes  infinite  for  x-=0.  Hence  the  denomi- 
nator of  the  transformed  fraction  will  be  ^   (putting  for 

brevity  I  =  log  x).  The  successive  derivatives  of  this  quantity 
with  respect  to  x  are 


xl 


— •     ^M      I   ^  +  ^V    etc 


The  successive  derivatives  of  the  numerator  are 

wa;"*"^;    m^ni  —  l)x^~^',    etc. 

The  limiting  values  of  the  given  quantity  x'^V*'  thus  become 
^r^min  +  i  m{m  —  l)x^ 


„m 


n 


n 


1    ,  ri  +  iy 

n  +  1     1       l^  +»  J 


etc.. 


which  remain  indeterminate  in  form  how  far  soever  we  may 
carry  them. 


'i! 


INDETERMINATE  F0RM8. 


133 


ihus  become 


ever  we  may 


In  such  cases  the  required  limit  of  the  fraction  can  be 
found  only  by  some  device  for  which  no  general  rule  can  bo 
laid  down.  In  the  example  just  given  the  device  consists  in 
replacing  a;  by  a  new  variable  y,  determined  by  the  equation 

log  a;  =  —  y. 
We  then  have  x  =  c~'". 

Since  for  x  i  0  y  i  oo ,  we  now  have  to  find  the  limit  of 

f  or  y  i  00 . 
By  taking  the  successive  derivatives  of  the  two  terms  of 

the  fraction  —-,  we  have  the  successive  forms 


ny""'  \    n{7i  —  l)y  "-^^    n{n  --  1)  {71  —  %)y 


n  — 3 


etc. 


Whatever  the  value  of  n,  we  must  ultimately  reach  an  ex- 
ponent in  the  numerator  which  shall  be  zero  or  negative,  and 
then  the  numerator  will  become  n\  if  n  is  a  jiositive  integer, 
and  will  vanish  for  y  =  co ,  if  71  is  not  a  positive  ^.iteger.  But 
the  denominator  will  remain  infinite.  Wo  therefore  con- 
clude: 

lim.  [a:'"(log  x^]  (x  ^  0)  =  0, 

whatever  be  m  and  71,  so  long  as  7n  i   positive. 

From  this  the  student  should  show,  by  putting  z~x~''-  and 
m  =  1,  that  the  fraction 

z 

becomes  infinite  with  z,  how  great  soever  the  exponent  71,  and 
therefore  that  any  iTiJinite  munber  is  an  infinity  of  higher 
order  than  any  power  of  its  logarithm, 

73.  Foi-m  00  —  00 .  In  this  case  we  have  an  expression  of 
the  form 

F{x)  =  u  —  V, 


.'  ''i 
>  ''P 
1(1 


I  ..' 


^V)\ 


J  ( 


134 


THE  DIFFEUIUNTIAL  CALCULUS. 


I.    I 


in  which  both  u  and  v  become  infinite  for  some  value  of  x. 
Placing  it  in  the  form 

F{x) = «(i  -  g, 

wo  see  that  F{j^  will  become  infinite  with  u  unless  the  fraction 

v 

—  approaches  unity  as  its  limit.     When  this  is  the  case  the 

expression  takes  the  form  qo  x  0  of  the  preceding  article. 

74.  Form  1".     To  investigate  this  form  lot  us  find  the 
limit  of  the  expression 

'i+ir 


(- 


nl 


u 


when  n  becomes  infinite.     Taking  the  logarithm,  we  have 
log  ?«  =  hn  log  ^1  +  -j 

=  hn  \ — -.  +  j7—,  —  ...!• 

Making  n  infinite,  we  have 

lim.  log  u  =  U; 

or,  because  the  limit  of  log  u  is  the  logarithm  of  lim.  u, 

log  lim,  11  =  h. 

lim.  (l  +  ^)''V:^oo)  =  e\ 


Hence 


In  order  that  this  result  may  be  finite,  h  itself  must  not  be 
infinite.     Wo  therefore  reach  the  general  conclusion: 
Theorem.    In  order  that  an  expression  of  the  form 

(1  +  ccY 

may  have  a  finite  limit  when  a  becomes  infinitesimal  and  x 
infinite,  the  product  ax  must  not  become  infinite. 

Cor,     If  the  product  ax  approaches  zero  as  a  limit,  the 
given  expression  will  approach  the  limit  unity. 


INDETERMINATE  FORMS. 


136 


75.  Forms  0*  and  oo*.  Lot  an  expression  taking  either 
of  these  forr  s  as  a  limit  be  represented  by  u^^F.  The 
problem  is  to  find  the  limiting  value  of  the  expression  when 
0  approaches  zero  and  i«  either  approaches  zero  or  becomes 
infinite. 

From  the  identity  w  =  c '««  " 

we  derive  F=ti'^  =  e^  •"«  ". 

We  infer  that  the  limit  of  F  will  depend  upon  that  of  <p  log  u. 

If  lim.  0  log  uiB  -\-  CO,  then  lim.  F=  ao. 

If  lim.  0  log  t*  is  —  00 ,  then  lim.  F=  0. 

If  lim.  0  log  ti  is  0,  then  lim.  F=  1. 

If  lim.  0  log  u  is  finite  then  lim.  F  is  finite. 
Hence  the  rule:    To  find  the  limit  of  m*  when  (p  :^0  and 
w  -  0  or  00 ,  put  I  =  lim.  0  log  u.    Then 

lim.  w*  =  eK 


ii 


imal  and  x 


EXAMPLES    AND    EXERCISES. 

1.  Find  lim.  of    for    x  =  0. 
Here  af  =  «*»««*. 

Since  x  log  x  has  zero  as  its  limit  when  a;  =  0,  the  required 
limit  is  e°  or  1. 

2.  lim.  a;"*    for    a;  i  0. 


3.  lim.  a:*    for    a;  ::!:  00. 

1 

4.  a:i-»    for    a;  =  1. 

tt 

5.  a;!-*    for    a;  =  1. 

h 

6.  (1  —  a;)*     for    a;  =  0. 


7. 


e''  —  ^-^ 


for    X  =  0. 


log  (1  +  a^) 

„    loff  sin  2x     . 

8.  -Y^ — -, —     for    a:  =  0. 
log  sm  X 

e*  +  log  (1  -  a;)  -  1 

9.  — ' -\ 

X  —  tan  X 


Ans,  F=l. 

Ans.  F=l. 

Ans.  — . 
e 

Ans.  e~\ 
Ans.  e~\ 
Ans.  2. 


Ans.  log  2. 
for    a;  =  0,    Ans,  J. 


Ii 


t!  I 


136 


THE  DIFFERENTIAL  CALCULUS. 


lo. 


m 


for    X  = 


00 


Ans,  1. 


II.  X  tan  X  — 


n 
2 


71 


sec  X    for    x  =  - 


12. 


13. 


14. 


15. 


<1! 


y  sin  -    for    y  = 


y 


00 


/tan  :z;y 

/tan  x"'x 
\   x    i 


for    »  =  00 


for    ;i;  =  0. 


for    a;  =  0. 


Ans,  —  1. 


^w 


•1;.  a. 


Ans.  log  a. 


-i/i5.  1. 


Ans.  ei, 


16.  (cos  x)7^    for    .r  =  0. 


7t 


17.  (1  -  y)  tan  -y    for 


2 


y 


[8. 


log  a:\ 


^    / 


for    X  =  0. 


19.  a;  —  a;"  log 


(^  -^  a 


I— « 


20. 


log  (1  +  x) 


for    X  =  0, 


Ans. 


-i 


Ans.  —. 
n 


Ans.  1. 


for     :c  =  00.      ^;i5.  |. 


/Iw5.  2. 


21, 


rt. 


+ 


^f. 


2 


for    a;  =  0. 


/lW5. 


rt.flj 


l^v,. 


22. 


(' 


'(^^"  +  «/+...+ 


a,^\^ 


n 


for  a:^0.    ^ 


ns.  a. a 


r'^a 


«, 


23.  Show  thtit,  how  great  soever  the  exponent 


n. 


X 


(log  xy 


~  00  when    X  ~  00 


j<7j  .  .  •  dn' 


flaws:  cubves. 


137 


CHAPTER  XI. 

OF   PLANE   CURVES. 

76.  Forms  of  the  Equations  of  Curves.  As  we  have  here- 
tofore considered  curve  lines,  they  have  heen  defined  by  an 
equation  between  the  co-ordinates  of  each  point  of  the  curve, 
and  therefore  of  one  of  the  forms 

y=/W;  ^=/(2/);  (i) 

and  F{x,  y)  =  0. 

The  distinguishing  feature  of  the  equation  is  that  when  we 
assign  a  vahie  at  pleasure  to  one  of  the  co-ordinates  x  or  y, 
one  or  more  corresponding  values  of  the  other  co-ordinate  are 
determined  by  the  equation. 

But  the  relation  between  x  and  y  may  be  equally  well 
defined  by  expressing  each  of  them  as  a  function  of  an 
auxiliary  variable,  which  is  then  the  independent  variable. 
Calling  this  auxiliary  variable  u,  the  equations  of  a  curve  will 
be  of  the  form 

y  =  Un).  f  ^^^ 

Assigning  values  at  pleasure  to  «,  we  shall  have  correspond- 
ing values  of  x  and  y  determining  each  point  of  the  curve. 

An  advantage  of  this  method  of  representation  is  that  for 
each  value  of  u  wo  have  one  definite  point  of  the  curve,  or 
several  definite  points  when  the  equations  give  several  values 
of  the  co-ordinates  for  each  value  of  Wy  and  wo  tlius  have  a 
relation  between  a  point  and  the  algebraic  quantity  u. 

It  is  also  to  be  remarked  that  by  eliminating  u  from  the 
efjuations  (3)  we  shall  get  a  single  equation  between  x  and  y 
which  will  be  the  equation  of  the  curve  in  one  of  the  forms 

(1). 


i  . 


W  : 


i!  1 


i  M 


i".;  1 


»  M 


i  ; 


138 


THE  DIFFERENTIAL  CALCULUS. 


I!  'I 


Example  1.     Let  us  put 

a,I}  =  the  co-ordinates  of  any  fixed  point  i?  of  a  straight  line; 

(X  =  the  angle  which 
the   line    makes  with         -J  «  E^ 

the  axis  of  x; 

p  =  the  distance  of 
any  point  P  of  the 
line  from  the  point 
(a,  h). 

Then  we  readily  see 
from  the  figure  that 
the  co-ordinates  x  and 
y  oi  P  are  given  by  the  equations 

X  =^  a  -{-  p  cos  a; 
y  z=])  -\-  p  sin  a 


Fio.  16. 


;! 


(3) 


which  are  equations  of  the  straight  line  in  the  independent 
form. 

Here  p  is  the  auxiliary  variable,  called  ?*  in  Eq.  3.     By 
eliminating  this  quantity  we  shall  have 

x^in  a  ~  y  cos  a  ^=  a  m\  a  —  h  cos  o', 
which  is  the  equation  of  the  line  in  one  of  its  usual  forms. 

Example  2.     The  equation  of  a  circle  may  be  expressed  in 

the  form 

X  z=  a  -{-  c  cos  n; 

y 


z=^  a  -{■  c  COB  u\  \ 
=  i  -f-  ^  sin  w;  ) 


(4) 


u  being  the  independent 
variable. 

By  writing  (4)  in  the 
form 

X  —  a=z  c  cos  u, 
y  —  h  =  c  sin  u, 
and    eliminating    n    by 
taking   the  sum  of   the 
squares  of  the  two  equa- 
tions, we  have 


Fio.  17. 


PLANE  CURVES. 


139 


(^  -  cty  -\-{y-  by  =  c\ 

the  equation  of  a  circle  of  radius  c. 

Notice  the  beautiful  relation  between  (3)  and  (4).  They 
are  the  same  in  form:  if  in  (4)  we  write  p  for  c  and  a  for 
u,  they  will  be  the  same  equations.  Then,  by  supposing  p 
constant  and  a  variable,  we  are  carried  round  '.he  point  (a,  h) 
at  a  constant  distance  p,  that  is,  around  a  circ  le.  By  suppos- 
ing p  variable  and  a  constant  we  are  carried  through  {a,  b) 
in  a  constant  direction,  that  is,  along  a  straight  line. 

77.  Infinitesimal  Elements  of  Curves.  Let  F  and  P'  be 
two  points  on  a  curve,  P  being  supposed 
fixed,  and  P'  variable.  We  may  then  sup- 
pose P'  to  approach  P  as  its  limit,  and  in- 
quire into  the  limits  of  any  magnitudes 
associated  with  the  curve. 

We  may  also  measure  the  length  of  an 
arc  of  the  curve  from  an  initial  point  G  to 
a  terminal  point  P.     Then,  supposing  C  fixed  and  P  variable, 
PP'  may  be  taken  as  an  increment  of  the  arc. 

If  we  put 

s  =  arc  CP, 

we  shall  have 

^s  =  arc  PP\ 

Axiom.  Tlie  ratio  of  an  infinitesimal  clement  of  a  curve 
to  the  straight  line  joining  its  extremities  approaches  unity  as 
its  limit. 

We  call  this  proposition  an  axiom  because  a  really  rigorous 
demonstration  does  not  seem  possible.  Its  truth  will  appear 
by  considering  that  if  the  curve  has  no  sharp  turns,  which 
we  presuppose,  then  it  can  change  its  direction  only  by  an  in- 
finitesimal quantity  in  any  infinitesimal  portion  of  its  length. 
Now,  a  line  which  has  the  same  direction  throughout  its  length 
is  a  straight  line. 


Fio.  18. 


'  !   i 
.if  ! 


■  '■  I ; 


\o 


Ml 


140 


THE  DIFFERENTIAL  CALCULUS. 


78.  Theorem  I.  If  a  straight  line  touch  a  curve  at  the 
point  F,  a  point  P'  on  the 
curve  at  an  infinitesimal 
distance  willy  in  general, 
be  distant  from  the  tangent 
hy  an  infinitesimal  of  the 
second  order. 

Let  y  =■  f  (x)  be  the  _o 
equation  of  the  curve.  fiq.  lo. 

Let  us  transform  the  equation  to  a  new  system  of  co-ordi- 
nates, x'  and  y' y  so  taken  that  the  axis  of  X'  shall  be  parallel 

dxi' 
to  the  taixgent  at  P.    This  will  make  y^  =  0.    Let  x'  and  y' 

be  the  co  ordinates  of  P,  and  {x'  -f-  h,  y")  the  co-ordinates 
of  a  point  P'  near  P, 
Developing  by  Taylor's  theorem,  we  have 


y 


ff 


y  ~dx'"^  dx''  1-2+ — 


i<    < 


Now,  y"  —  y'  is  the  distance  P'Q  of  the  point  P'  from  the 

dy' 
tangent  at  P.     Since  ~  =  0,  when  h  becomes  infinitesimal 

the  term  of  highest  order  in  this  distance  is  -~  —-,  a  quan- 

ax     X  i\) 

tity  of  the  second  order. 

Remark.  In  the  special  case  when  —-p^  =  0,  the  distance 

in  question  may  be  a  quantity  of  the  third  or  of  some  higher 
order,  according  to  the  order  of  the  first  differential  coeffi- 
cient which  does  not  vanish. 

Corollary.  The  cosine  of  an  infinitesi^nal  arc  differs 
from  unity  hy  an  ififinitesimal  of  the  second  order. 

For  if  wc  draw  a  unit  circle  with  its  tangent  at  the  initial 
point,  the  cosine  of  an  arc  will  differ  from  unity  by  the  dis- 
tance from  the  end  of  the  arc  to  the  tangent  line.  AVhen  tlie 
arc  is  infinitesimal,  the  corollary  follows  from  the  theorem. 


PLANE  CTfRVE8. 


141 


le  distance 


Theorem  II.  The  area  included  between  an  infinitesimal 
arc  and  its  cliord  is  not  greater  than  an  infinitesimal  of  the 
tk''^d  order. 

From  Th.  I.  we  may  readily  see  that  the  maximum  distance 
between  the  chord  and  its  arc  is  a  quantity  of  the  second 
order.  The  area  is  less  than  the  product  of  this  distance  by 
tlie  length  of  the  chord,  which  product  is  an  infinitesimal  of 
at  least  the  third  order. 

•79,  Expressions  for  Elements  of  Curves.  Def,  An 
element  of  a  geometric  magnitude  is  an  infinitesimal  por- 
tion of  that  magnitude. 

The  wore;  implies  that  we  conceive  the  magnitude  to  be 
made  up  of  infinitesimal  parts. 

Elemefit  of  an  Arc,     Let  us  put 

s  =  the  length  of  any  arc  of  a  curve; 
ds  E  an  element  of  this  arc. 

If  F  and  P'  be  two  points  of  a  curve,  we  shall  have 

(chord  PPy  =  Ax"  +  Ay\ 

When    PP'  becomes    infinitesimal,  as 

the  ratio  of  ds  to  PP'  becomes  unity 
(§  77),  and  we  have  y-        Aa; 

ds"^  =  dx^  -\-  d]f\ 

ds  =  *^dx'  +  dif  =  yi-\-  i-?S(^^' 

Case  of  Polar  Co-ordinates.  To  express  the  element  of  a 
curve  referred  to  polar  co-ordinates,  differentiate  the  equa- 
tions 

X  =  r  cos  0;    y  =  r  sin  0. 

Thus  dx  =  cos  0dr  —  r  sin  0d0; 

dy  =  sin  ddr  -f-  r  cos  BdO', 
which  gives  ds"^  =  dr^  -f  r^dff^ 


Fio.  20. 


M;1 


i  H 


'1 

'    ■II 


/■•■■ 


\ 


I , 


i 


iiil 


and 


.u  =  \^^  +  [-)  ae. 


m 


1 

■    >l 


143 


THE  DIFFERENTIAL  CALCULUS. 


80.  Equations  of  certain  Noteworthy  Curves.  The  Cycloid. 
The  cycloid  is  a  curve  described  by  a  point  on  the  circumfer- 
ence of  a  circle  rolling  on  a  straight  lin*^.  A  point  on  the 
circumference  of  a  carriage-wheel,  as  the  carriage  moves, 
desciibes  a  series  of  cycloids,  one  for  each  revolution  of  the 
wheel. 

To  find  the  equation  of  the  cycloid,  let  P  be  the  generating 
point.  Let  us  take  the  line  on  which  the  circle  rolls  as  the 
axis  of  X,  and  let  us  place  the  origin  at  the  point  0  where  P 
is  in  contact  with  the  line  OX. 


!     ' 


O  Q  K  B      ^ 

Fia.si. 

Also  put 

a  E  the  radius  of  the  circle ; 

u  E  the  angle  through  which  the  circle  has  rolled,  expressed 
in  terms  of  unit-radius. 

Then,  when  the  circle  has  rolled  through  any  distance  OR, 

this  distance  will  be  equal  to  the  length  of  the  arc  PR  of  the 

circle  between  P  and  the  point  of  contact  R,  that  is,  to  au. 

We  thus  have,  for  the  co-ordinates  of  the  centre,  G,  of  the 

circle, 

z  =  au'y 

y^a\ 
and  for  the  co-ordinates  of  the  point  P  on  the  cycloid. 


x  =  aw  —  fl  sin  w  =  a(\i  —  sin  w); 
y  =  rt    —  fl  cos  w  =  rt(l  —  cos  u) 


;1 


(1) 


which  an  the  equations  of  the  cycloid  with  w  as  an  independ- 
ent variable. 


PLANE  CURVES. 


143 


To  eliminate  u,  find  its  value  from  the  second  equation^ 

w  =  cos<-«(l-^V 

This  gives 


sin  u  =  ^1  —  cos'  u  = — ^. 

a 


Then,  by  substituting  in  the  first  equation 

X  =  a  cos<~** —  V2ay  —  y% 

which  is  the  equation  of  the  cycloid  in  the  usual  form. 


(2) 


81.  The  Lemniscate  is  the  locus  of  a  point,  the  product  of 
whose  distances  from  two  fixed  points  (called  foci)  is  equal 
to  the  square  of  half  the  distance  between  the  foci. 

Let  us  take  the  line  joining  the  foci  as  the  axis  of  X,  and 
the  middle  point  of  the  segment  between  the  foci  as  the 
origin.  Let  us  also  put  cEhalf  the  distance  between  the 
foci. 


Fio.  22. 


Then  the  distances  of  any  point  (x,  y)  of  the  curve  from 
the  foci  are 


.      V(x-cy-{-y^     and     V(x  +  cy-\-y\ 

Equating  the  product  of  these  distances  to  c',  squaring  and 
reducing,  we  find 

(x^  +  yy  =  2c'(a:'  -  y%  (3) 

which  is  the  equation  of  the  lemniscate. 


'I      ( 

'I  '■ 

--'h 


I 


M 


* 


H 


t 


144 


J' 

i! 


ill 


11 


fr 


T5:fi?  DIFFERENTIAL  CALCULUS. 


(5) 


Transforming  to  polar  co-ordinates  by  the  substitutions 

X  =  r  cos  6, 
y  =  r  sin  ^, 
we  find,  for  the  polar  equation  of  the  lemniscate, 

r'  =  2c»  cos  2^.  (4) 

Putting  ?/  =  0,  we  find,  for  the  point  in  which  the  curve 
cuts  the  line  joining  the  foci, 

x=  ±  i^2c  =  a. 
The  line  a  is  the  semi-axis  of  the  lemniscate.     Substitut- 
ing it  instead  of  c,  the  rectangular  and  polar  equations  of  the 
curve  will  become 

{X' +  fy  =  a\x^  -  f); 
r^  =  a^  cos  2/9. 

83.  The  Arcliimedean  Spiral.  This  curve  is  generated 
by  the  uniform  motion  of  a  point  along  a  line  revolving  uni- 
formly about  a  fixed  point. 

To  find  its  polar  equation,  let  us  take  the  fixed  point  as  the 
pole,  and  the  position  of  the  revolving  line  when  the  generat- 
ing point  leaves  the  pole 
as  the  axis  of  reference. 
Let  us  also  put 

a  E  the  distance  by 
which  the  generating 
point  moves  along  the 
radius  vector  \/hile  the 
latter  is  turning  thr'^^igh 
the  unit  radius. 

Then,  when  the  ra- 
dius vector  has  turned 
through  the  angle  6,  the 
point  will  have  moved 
from  tlie  pole  through  the  distance  aO, 

r  =  aO 


Fig.  23. 

Hence  we  shall  have 


as  the  polar  equation  of  the  Archimedean  spiral. 


PLANE  GUnVES. 


145 


If  we  increase  6  by  an  entire  revolution  (27r),  the  corre- 
sponding increment  of  r  will  be  llTia,  a  constant.     Hence: 

The  Archimedean  spiral  cuts  any  fixed  jmsitioii  of  the  ra- 
dius  vector  in  an  indefinite  series  of  eqiiidistant  points, 

83.  The  Logarithmic  Spiral.  This  is  a  spiral  in  which 
the  logarithm  of  the  radius  vector  is  proportional  to  the  angle 
through  which  the  radius  vector  has  moved  from  an  initial 
position.  Hence,  if  we  put  6^ 
for  the  initial  angle,  we  have 

log  r  =  l{e-  0X 
I  being  a  constant.     Hence 

19  -  19,  -  Wo  10 

r  =  e        *  =  e      "e  , 
Putting,  for  brevity, 

a=ze       , 

the  equation  of  the  logarith- 
mic spiral  becomes  fio.  24. 

r  =  ae^^, 
a  and  I  being  constants. 

EXERCISES. 

1.  Show  (1)  that  the  maximum  ordinate  of  the  lemniscate 
is  |c,  and  (3)  that  the  circle  whose  diameter  is  the  line  join- 
ing the  foci  cuts  the  lemniscate  at  the  points  whose  ordinatea 
are  a  maximum. 

2.  Find  the  following  expression  for  the  square  of  the  dis- 
tance of  a  point  of  a  cycloid  from  the  starting  point  (0,  Fig. 

21): 

r  =  2ay  -{-  'Huax  —  a'w*. 

3.  A  wheel  makes  one  revolution  a  second  around  a  fixed 
axis,  and  an  insect  on  one  of  the  spokes  crawls  from  the  cen- 
tre toward  the  circumference  at  the  rate  of  one  inch  a  second. 
Find  the  equation  of  the  spiral  along  which  he  is  carried. 


^1 


'  1 


t 


I ' 


t 
'^1 


146 


THE  DIFFERENTIAL  CALCULUS. 


4.  If,  in  that  logarithmic  spiral  for  which  ^  =  1  and  1=1, 


r  =  6*, 


the  radius  Toctor  turns  through  an  arc  equal  to  log  2,  its 
length  will  be  doubled. 

5.  li,  in  any  logarithmic  spiral,  one  radius  vector  bisects 
the  angle  between  two  others,  show  that  it  is  a  mean  propor- 
tional between  them. 

6.  Show  that  the  pair  of  equations 


X  =  au , 
represent  a  parabola  whose  parameter  is 


a' 


7.  If,  in  the  equation  of  the  Archimedean  spiral,  0  and 
therefore  r  take  all  negative  values,  show  that  we  shall 
have  another  Archimedean  spiral  intersecting  the  spiral  given 
by  positive  values  of  0  in  a  series  of  points  lying  on  a  line  at 
right  angles  to  the  initial  position  of  the  revolving  line. 

This  should  be  done  in  two  ways.  Firstly,  by  drawing  the  continua- 
tion of  the  spiral  when,  by  a  negative  rotation  of  the  revolving  line,  the 
generating  point  passes  through  the  pole.  It  will  then  be  seen  that  the 
combination  of  the  two  spirals  is  symmetrical  with  respect  to  the  vertical 
axis.  Secondly,  by  expressing  the  rectangular  co-ordinates  of  a  point  of 
the  spiral  in  terms  of  0  we  have 

«  =  aG  cos  0, 
y  =  aO  sin  0. 

Changing  the  sign  of  0  in  this  equation  will  change  the  sign  of  x  and 
leave  y  unchanged. 

8.  Show  that  if  we  draw  two  lines  through  the  centre  of  a 
lemniscate  making  angles  of  45°  with  the  axes,  no  point  of 
the  curve  will  be  contained  between  these  lines  and  the  axis 
of  V. 


n  of  0!  and 


TANGENTS  AND  NORMALS. 


147 


CHAPTER  XII. 

TANGENTS   AND    NORMALS. 

84.  A  tangent  to  a  curve  is  a  straight  liii^  through  two 
coincident  points  of  the  curve. 


Fia.  25. 

A  normal  is  a  straight  line  through  a  point  of  the  curve 
perpendicular  to  the  tangent  at  that  point. 

The  subtangent  is  the  projection,  TQy  upon  the  axis  of 
X,  of  that  segment  TP  of  the  tangent  contained  between 
the  point  of  contact  and  the  axis  of  X, 

The  subnormal  is  the  corresponding  projection,  QN,  of 
the  segment  PN  of  the  normal. 

Notice  that  a  tangent  and  a  normal  are  linos  of  indefinite 
length,  while  the  subtangent  and  subnormal  are  segments  of 
the  axis  of  abscissas.  Hence  the  former  are  determined  by 
their  equations,  which  will  be  of  the  first  degree  in  x  and  y, 
while  the  latter  are  determined  by  algebraic  expressions  for 
their  length. 

But  the  segments  TP  and  PN"  are  sometimes  taken  as 
lengths  of  the  tangent  and  normal  respectively,  when  we  con- 
sider these  lines  as  segments. 


M 

i 


t 


1  4jl 


I. 'I 


t 
II 


3    i 


1  ' 


i'S 


! 


} 


148 


y//^  DIFFERENTIAL  CALCULUS. 


85.  General  Eqiiat ion  for  n  Tdwient.  The  general  prob- 
lem of  tangents  to  a  curve  may  bo  stated  th«s: 

7b  find  the  conditvm  which  the  parameters  of  a  straight 
line  must  satisfy  in  order  that  the  line  may  be  tangent  to  a 
given  cnrve. 

But  it  is  commonly  considered  in  the  more  restricted  form: 
To  find  the  equation  of  a  tangent  to  a  curve  at  a  given  point  on 
the  curve. 

Ijct  {x^f  ?/,)  be  the  given  point  on  the  curve.  By  Analytic 
Geometry  the  equation  of  any  straight  line  through  this  point 
may  be  expressed  in  the  form 

y-y,-m{x-  xy,  (5) 

m  being  the  tangent  of  the  angle  which  the  line  makes  with 
the  axis  of  X,     But  we  have  shown  (§  30)  that 

*"  -  dx: 

this  differential  coefficient  being  formed  by  differentiating  the 
equation  of  the  curve.     Uenco 


(6) 


is  the  equation  of  the  tangent  to  any  curve  at  a  point  (a;,,  y,) 
on  the  curve. 

Equation  of  the  Normal.  The  normal  at  the  point  (a;,,  y,) 
passes  through  this  point,  and  is  perpendicular  to  the  tangent. 
If  m'  be  its  slope,  the  condition  that  it  shall  be  perpendicular 
to  the  tangent  is  (An.  Geom. ) 

—  =  _  JL 

m  ~       dyl 
dx. 


m'  = =  - 


Hence  the  equation  of  the  normal  at  the  point  (a:,,  y,)  is 


dx 


iy  -  ?/,)  =  a;, 


x. 


m 


TANGENTS  AND  N0UMAL8, 


149 


In  these  equations  of  the  tangent  and  normal  it  is  necoBsary 
to  distinguish  between  the  cases  in  which  the  symbols  x  and 
y  represent  the  co-ordinates  of  points  on  the  tangent  or  nor- 
mal line,  and  those  where  they  represent  the  given  point  of 
the  curve.  Where  both  enter  into  the  same  equation,  one  set, 
that  pertainiag  to  the  curve^  must  be  marked  by  suflixes  or 
accents. 

86.  8%ibtangent  and  Subnormal.  To  find  the  length  of 
the  subtangent  and  subnormal,  we  have  to  find  the  abscissa 
a;,  of  the  point  T  in  which  the  tangent  cuts  the  axis  of  abscis- 
sas.    We  then  have,  by  definition, 


Fio.   26. 

Subtangent  =  x^  —  x^ 

The  value  of  x^  is  found  by  putting  y  =  0  and  x 
the  equation  of  the  tangent.     Thus,  (6)  gives 

-  ^' = ^;(^«  -  ''■>• 

Hence,  for  the  length  of  the  subtanorent  TQ, 
Subtangent  =  a;,  —  a;„  =  J^, 

We  find  in  the  same  way  from  (7),  for  QN, 

Subnormal  =  —  t/.^'« 

dx^ 


x^m 


(8) 


(9) 


i    ' 


P 


ii» 


1  jjlH!' 


«  ; 

fault         * 

ii 

H 

!    : 

\ ' 

•}\ 

I 


160 


TEE  DIFFERENTIAL  CALCULUS. 


:i  :■ 


:i     5' 


87.  Modified  Forms  of  the  Equation.  In  the  preceding 
discussion  it  is  assumed  that  the  equation  of  the  curve  is  given 
in  the  form 

But,  firstly,  it  may  be  given  in  the  form 

F{x,  y)  =  0. 

We  shall  then  have  (§  37) 

dF 
dy^  dx^ 

Substituting  this  value  in  the  equations  (6)  and  (7),  we  find 


^  .    dF,  .       dF.  .    "i 

Tangent:  -^(y  -  yj  =  ^{x,  -  x);  • 

Normal:    -y~  (i/  —  V,)  =  ^— («  —  «,). 
dz,  -       ^*'      dy^^  " 


(10) 


Sec  adly;,  if  the  curve  is  defined  by  two  equations  of  the 
form 


(11) 


we  have 


dy^  __  du 
dx^  ~  dx ' 
du 


in  which  there  is  no  need  of  suffixes  to  x  and  y  in  the  second 
member,  because  this  member  is  a  function  of  u,  which  does 
not  contain  x  or  y. 
By  substitution  in  (6)  and  (7),  we  find 


Eq.  of  tangent:  {y  -  y,)^  =  {x  -  a:,)^.  ' 


Eq.  of  normal:    (y  -  y,)'^  =  (a;,  -  x)^. 


du 


y  (13) 


TANGENTS  AND  NORMALS. 


151 


By  substituting  in  these  equations  for  a;,y„  -r-  and     ^ 

ay 


du 


their  values  in  terms  of  u,  the  parameters  of  the  lines  will  be 
functions  of  w.  Then,  for  each  value  we  assign  to  u,  (11) 
will  give  the  co-ordinates  of  a  point  on  the  curve,  and  (12) 
will  determine  the  tangent  and  normal  at  that  point. 

88.  Tangents  and  Normals  to  the  Conic  Sections.  Writing 
the  equation  of  the  ellipse  in  the  form 

oY  4-  *•«•  =  a^h\  (a) 

we  readily  find,  by  differentiation, 

dy  _      b*x 
dx  ~~      a*y' 

Applying  the  suffix  to  x  and  y,  to  show  that  they  represent 
co-ordinates  of  points  on  the  ellipse,  substituting  in  (6)  and 
(7),  and  noting  that  x^  and  y,  satisfy  (a),  we  readily  find: 

For  the  tangent:  ^  +  ^tt  =  1. 

a*        h* 
For  the  normal:    -x y  —  a*  —  h*. 

Taking  the  equation  of  the  hyperbola, 

we  find,  in  the  same  way. 

For  the  tangent:  ^  —  ^  =  1. 

a"        h* 
For  the  normal:    -xA —  y  =  a'  4-  5". 

Tr-king  the  equation  of  the  parabola, 

we  find,  by  a  similar  process. 

For  the  tangent:        y,y  =  p(x  -f  a;,). 

For  the  normal:   y  ^  y^:=  ^  -{^i  —  ^)» 


I:    ^1;' 


<-,: 


:  i 


iih 


i  t 

■A 


■  i 


s  -i 


SI 


','"' 


5 


162 


THE  DIFFERENTIAL  CALCULUS. 


89.  Problem.     To  find  the  length  of  the  perpendicular 
dropped  from  the  origin  upon  a  tangent  or  normal. 

It  is  shown  in  Analytic  Geometry  that  if  the  equation  of  a 
straight  line  be  reduced  to  the  form 

Ax-\-By-\-G=0, 
the  perpendicular  upon  the  line  from  the  origin  is 

C 


VA'  +  B' 

It  must  be  noted  that  in  the  above  form  the  symbol  C  rep- 
resents the  sum  of  all  the  terms  of  the  equation  of  the  line 
which  do  not  contain  either  x  or  y. 

If  we  have  the  equation  of  the  line  in  the  form 

y-y,  =  m{x-  x^),  ^ 

we  write  it  mx  —  y  —  mx^  -f  y^  =  0,  .     ' 

and  then  we  have 

-4  =  m; 

0=y,-  mx^. 
Thus,  the  expression  for  the  perpendicular  is 

y,  —  mx, 
p  =  ^'  '- 

Vw'  + 1 
Substituting  for  m  the  values  already  found  for  the  tan- 
gent and  normal  respectively,  we  find. 
For  the  perpendicular  on  the  tangent : 


P 


/ 


^  +  4)' 


ds 


(1) 


For  the  per2)endicular  on  the  normal : 


x4-V^' 


p  = 


^^+(S:)' 


ds 


(2) 


TANGENTS  AND  NORMALS. 


163 


Fig.  27. 


90.  Tangent  and  Normal  in  Polar  Co-ordinates, 

Problem.  To  find  the 
angle  which  the  tangent  at 
any  point  makes  with  the 
radius  vector  of  that  point. 

Let  PP'  be  a  small  arc 
of  a  curve  referred  to  polar 
co-ordinates; 

KP,  a  small  part  of  the 
radius  vector  of  the  point 
P  (the  pole  being  too  far 
to  the  left  to  be  shown  in 
the  figure); 

K'P',  the  same  for  the  point  P\ 

KSR,  a  parallel  to  the  axis  of  reference.  Drop  PQlK'P'. 

Let  SPT  be  the  tangent  at  P,     We  also  put 
y  =  angle  KPS  which  the  tangent   makes  with  the  radius 
vector. 

Then  let  P'  approach  P  as  its  limit.     Then 

QP'  =  dr;     PQ  =  rdd; 
PQ    .   rdS 

^^^y^-QP^-Tr- 

We  also  have 

1  1  dr 


(1) 


cos  y 


4/(1  +  tan"  y) 


/k+en* 


sin  y  =  cos  y  tan  y  = 


i/|-^(l)T 


\  (3) 


Cor,  The  angle  RSP  which  the  tangent  makes  with  the 
^xis  of  reference  is  ^  -j-  6*. 


■  I 

X-. 


l[  ill 


t 


m 


» 


l!^    ! 


I'" 

m  1 


:i 


n->ii 


th    ' 


III 


164 


THE  DIFFERENTIAL  CALCULUS. 


91.  Perpendimdar  from  the  Pole  upon  the  Tangent  and 
Normal.  When  y  is  the  angle  between  the  tangent  and  the 
radius  vector,  we  readily  find,  by  geometrical  construction, 
that  the  perpenuicular  from  the  pole  upon  the  tangent  and 
normal  are,  respectively, 

j3  =  r  sin  ^^    and    p  —  r  cos  y. 

Substituting  for  sin  y  and  cos  y  the  values  already  found, 
we  have. 

For  the  perpendicular  on  tangent : 


p  = 


/l^'+ST 


For  the  jwrpendicular  on  normal : 

r 


P  = 


dr 


v^i^'^m"' 


(3) 


92,  Problem.  7b  find  the  equation  of  the  tangent  and 
normal  at  a  given  poi7it  of  a  curve  -whose  :'quatio7i  is  expressed 
in  polar  co-ordinates. 

It  is  shown  in  Analytic  Geome(.ry  that  if  we  put 

p  =  the  perpendicular  dropped  from  the  origin  upon  a  line; 
a  E  the  angle  which  this  perpendicular  makes  with  the 
axis  of  X; 

the  equation  of  the  line  may  be  written 

X  cos  fx  -}-  y  sin  a  —  p  =  0.  (1) 

Now,  as  just  shown,  the  tangent  makes  the  angle  y  -\-  (^ 
with  the  axis  of  X,  and  the  perpendicular  dropped  upon  it 
makes  an  angle  90°  less  than  this.     Hence  we  have 

a  =  y  ^6-  90°; 
cos  a  =       sin  (y  -{-  0)  =       sin  y  cos  d  -f-  cos  y  sin  6; 
sin  «  =  —  cos  {y  -{-  6)  =:  —  cos  y  cos  ^  +  sin  ;^  sin  d. 


TANGENTS  AND  NORMALS. 


155 


By  substitution  in  (1),  the  equation  of  the  tangent  becomes 

a:(8in  y  cos  0  +  cos  y  sin  6) 

—  y(coB  y  cos  0  —  Bin  y  sin  6)  —  p  =  0. 

Substituting  f o-  cos  y,  sin  y  and  ji  the  values  already  found, 
this  equation  of  the  tangent  reduces  to 

Ir  cos  &  -\-  -,n  sin  0jx-{-lr  sin  0  —  -jh  cos  6)  y  —  r*  =  0,  (2) 

r  and  0  being  the  co-ordinates  of  the  point  of  tangency. 

In  the  ease  of  the  normal  the  perpendicular  upon  it  is 
parallel  to  the  tangent.  Therefore,  to  find  the  equation  of 
the  normal,  we  must  put  in  (1) 

a  =  y  -\-  0, 

Substituting  this  value  of  a,  and  proceeding  as  in  the  case 
of  the  tangent,  we  find,  for  the  normal, 

-j^ cos  0  —  r  sin  0jc  -\-ir  cos  ^  +  ;t^ sin  ^J y  —  r-r-n  =  O*     (3) 

Generally  these  equations  will  be  more  convenient  in  use  if 
we  divide  them  throughout  by  r.     Thus  we  have: 

Equaiion  of  the  tangent : 
cos  ^  +  - ~  sin  0jx-\-  [sm  ^  -  - ^^ cos  0\y  -  r  =  0.    (4) 

Equation  of  the  normal  : 
1  dr 


rdS 


cos 


0-Bm0jx^  [^  ^~  sin  0  +  cos  0jy  -  ~  =  0.  (5) 


I 


In  using  these  equations  it  must  be  noticed  that  the  co- 
efficients of  X  and  y  are  functions  of  r  and  0,  the  polar  co- 


ar 


ordinates  of  the  point  of  tangency.     When  r,  0  and  -^^  are 

given,  this  point  and  the  tangent  through  it  are  completely 
determined. 


i 


W  il 


166 


a: 

ii 


?      ? 
I      f 

i'i 
ill 


Hi  i 


i 


THE  DIFFEBEyTlAL  CALCULUS. 


EXERCISES. 


1.  Show  that  in  the  case  of  the  Archimedean  spiral  the 
general  expressions  for  the  perpendiculars  from  the  pole  upon 
the  tangent  and  normal,  respectively,  are 

Thence  define  at  what  point  of  the  spiral  the  radius  vector 
makes  angles  of  45°  with  the  tangent  and  normal.  Find  also 
what  limit  the  perpendicular  upon  the  normal  approaches 
as  the  folds  of  the  spiral  are  continued  out  to  infinity. 

Show  also  from  §  92  that  the  tangent  is  perpendicular  to 
the  line  of  reference  at  every  point  for  which 

r  sin  (^  —  a  cos  6^  =  0, 

and  hence  that,  as  the  folds  of  the  spiral  are  traced  out  to 
infinity,  the  ordinates  of  the  points  of  contact  of  such  a  tan- 
gent approach  ±  a  as  their  limit. 

2.  Show  by  Eq.  12  that  in  the  case  of  the  logarithmic 
spiral  the  angle  which  the  radius  vector  makes  with  the  tan- 
gent is  a  constant,  given  by  the  equation 

tan  y  =-j-» 

3.  Show  from  Eq.  12  that  if  a  curve  passes  through  the 

pole,   the  tangent  at  that  point  coincides  with  the  radius 

dr 
vector,  unless  -^^  =  0  at  this  point.    Thence  show  that  in  the 

lemniscate  the  tangents  at  the  origin  each  cut  the  axes  at 
angles  of  45°. 

4.  Show  that  the  double  area  of  the  triangle  formed  by  a 

tangent  to  an  ellipse  and  its  axes  is .    Then  show  that  the 

area  is  a  maximum  when  — '  =  ±  p, 

a  0 

Show  also  that  the  area  of  the  triangle  formed  by  a  nor- 
mal and  the  axes  is  a  maximum  for  the  same  point. 


ASYMPTOTES  AND  SINGULAR  POINTS. 


157 


*?> 


CHAPTER  XIII. 

OF  ASYMPTOTES,  SINGULAR  POINTS  AND 
CURVE-TRACING. 

93.  Asymptotes.  An  asymptote  of  a  curve  is  the  limit 
which  the  tangent  approaches  when  the  point  of  contact  re- 
cedes to  infinity. 

In  order  that  a  curve  may  have  a  real  asymptote,  it  must 
extend  to  infinity,  and  the  perpendicular  from  the  origin  upon 
the  tangent  must  then  approach  a  finite  limit. 

For  the  first  ''ondition  it  sufiices  to  show  that  to  an  infi- 
nite value  cf  one  co-ordinate  corresponds  a  real  value,  finite 
or  infinite,  of  the  other. 

For  the  second  condition  it  suffices  to  show  that  the  expres- 
sion for  the  perpendicular  upon  the  tangent  (§§  89,  91)  ap- 
proaches a  finite  limit  when  one  co-ordinate  of  the  point  of 
contact  becomes  infinite.  If,  as  will  generally  be  most  con- 
venient, the  equation  of  the  curve  is  written  in  the  form 

F{x,y)  =  0,  (1) 

the  value  (1)  of  the  perpendicular,  omitting  suffixes,  may  be 
reduced  to 


P  = 


dF       dF 
^  dy         dx 


m 


+ 


\dv  /   ) 


(2) 


If  this  expression  approaches  a  real  finite  limit  for  an 
infinite  value  of  x  or  y,  the  curve  has  an  asymptote. 

If  the  curve  is  referred  to  polar  co-ordinates,  we  use  the 
expression  (3),  §  91,  for  j).  If  this  approaches  a  real  finite 
limit  for  an  infinite  value  of  r,  the  curve  has  an  asymptote. 


U  !ll 


Til 


H 


m 


:'t' 


I 


168 


TEE  DIFFERENTIAL  CALCULUS. 


% 


I   is 

Lis 

m 

% 


li 


m 


I 


The  existence  of  the  asymptote  being  thus  established,  its 
equation  may  generally  be  found  from  the  form  (10),  §  87, 
which  we  may  write  thus: 


dF     ,   dF  dF  ,      dF 


(8) 


by  supposing  ar,  or  y,  to  become  infinite. 

fl  F         (IF 
CommoD^     ^^)o  coefficients  -7-  and  -j—  will  them  selves  bo- 

dx^         dy, 

come  infinite   .p^itli    b^  co-ordinates.     We  must  then  divide 

the  whole  equation  by  such  powei  s  of  x^  and  y^  that  none  of 

the  terms  shall  become  infinite. 

94.  Examples  of  Asymptotes, 

1.  F(x)  =  x'-\-  y*—  daxy  =  O.(rt) 

The  curve  represented  by  this 
equation  is  called  the  Folium  of 
Descartes.  The  equation  (3)  gives 
in  this  case,  applying  suffixes, 

(x*  -  ay,)x  +  (y/  -  ax;)y 

=  «.'  +  y/  -  2aa;,y,  =  ax^y,. 
To  make  the  coefficients  of  x  and 
y  finite  for  a;^  =  00 ,  divide  by  a;,y,.    Then  the  equation  bo- 
comes 


Fio.  28. 


fx.      a\     ,  fy,      a  \ 

(-' ]x-{-  •-* ]y 

\y,      xj    ^W,      y,r 


a  =  0. 


W 


Let  us  now  find  from  (a)  the  limit  of  y^  for  a;,  =  00 .  Wo 
have 

X  X 

The  second  member  of  this  equation  will  approach  zero  as 
a  limit,  unless  y,  is  an  infinite  of  as  high  an  order  as  x*, 
which  is  impossible,  because  then  the  first  member  of  the 
equation  containing  y,'  would  be  an  infinite  of  higher  order 


ASYMPTOTES  AND  SINGULAR  POINTS 


169 


than  the  second  member^  which  is  absurd.    Hence,  passing 
to  the  limit, 

Urn.  (|')(rt,i=o)  =  -l. 

Then,  by  substitution  in  (b),  we  find,  for  the  asymptote, 

X -\-  y  ■\-  a  =■  0. 
2.  Take  next  the  equation 

F{x,  y)  =  x^  —  2x*tf  —  ax^  —  «'y  =  0.  (a) 

With  this  equation  (3)  becomes 

=  Zx*  —  6a;/y,  —  3a;     -  a'j/,.     (*) 


Fig.  29. 

"We  notice  that  the  terms  of  highest  order  in  the  second 

member  are  three  times  those  of  highest  order  in  (a).     From 

{a)  we  have 

X*  -  2x^'y  ~  ax*  +  a'y,. 

Substituting  in  tlie  second  member  of  (b),  and  dividing  by 
a:,*,  (/>)  becomes 

Solving  (a)  for  y,  we  find 

h  -  ^.'  -  ^^i 

an  expression  which  approaches  the  limit  ^  when  a;,  =  oo , 

Thus,  passing  to  the  limit,  (b')  gives,  for  the  equation  of  the 

asymptote, 

35  —  2y  =  a. 


|!» 


.il5 


!ii!il 


160 


THE  DIFFERENTIAL  CALCULUS. 


I 


I;  I  I 

ll 


i  -      ■ 


3.  The  Witch  of  Agnesi.    This  curve  is  named  after  the 
Italian  lady  who  first  investigated  its 
properties.     Its  equation  is 

x'y  +  a'y  —  a*  =  0.  {a) 

The  equation  of  the  tangent  is 

2x^y^x  +  (:c;  +  a')y  =  Sx^y,  +  a\  =  3a'  -  2a\.     (b) 

By  solving  (a)  for  x  and  y  respectively  we  see  that  a;,  may 

become  infinite,  but  that  y,  is  always  positive  and  less  than  a. 

Hence,  to  make  the  coefficient  of  y  in  (b)  finite  for  :r,  =  oo , 

we  must  divide  by  a:,',  which  reduces  the  equation  of  the 

asymptote  to 

y  =  0. 

Hence  the  axis  of  x  is  itself  an  asymptote.  \ 

95,  Points  of  Inflection.     A  point  of  inflection  is  a  point 
where  the  tangent  inter- 
sects   the    curve    at    the 
point  of  tangency. 

It  is  evident  from  the 
figure    that    in     passing 
along  the  curve,  and  con- 
sidering the  slope  of  the  ^^<*-  ^*- 
tangent  at  each  point,  the  point  of  inflection  is  one  at  which 
this  slope  is  a  maximum  or  a  minimum.     Because  we  have 

slope  =  |., 

the  conditions  that  the  slope  shall  be  a  maximum  or  minimum 
are 


dx*      ^ 


d'y 


and  -—  different  from  zero.     If  the  first  condition  is  fulfilled, 
but  if  -T^,  is  also  zero,  we  must  proceed,  as  in  problems  of  maxi- 


I  s! 


ASYMPTOTES  AND  SINGULAR  POINTS. 


161 


ma  and  minima,  to  find  tlie  first  derivative  in  order  which 
does  not  vanish.     If  the  order  of  this  derivative  is  even,  there 

is  no  point  of  inflection  for  -j^  =  ^J  if  odd,  there  is  one. 

As  an  example,  let  it  be  required  to  find  the  points  of  in- 
flection of  the  curve 

xy^  =  a'{a  —  x). 
Eeducing  the  equation  to  the  form 


we  find 


f 

~  X 

-a\ 

dy 
dx 

=  — 

a* 
2x'y' 

cry 
dx'~~ 

a* 

2x*v 

ii^^y  + 

^,dy> 
dx  J 

""  2x*v 

The  condition  that  this  expression  shall  vanish  is 

4:xy^  =  «*, 

which,  compared  with  the  equation  of  the  curve,  gives,  for  the 
co-ordinates  of  the  point  of  inflection. 


x^-^a;    y 


V'6 


EXERCISES. 


Find  the  points  of  inflection  of  the  following  curves  : 


X 


I.   xy  =  rt'  log  -. 


Ans. 


X  =  ac^. 


a 


ix  =  a{l  —  cos  u); 


y 


|«e    *. 


2.     1'^  =  ''^ 

\y  =  a{nu  +  sin  u). 


Ans.   < 


X 


n 


,=4o.-..(-L)+i^). 


,  I. 


.^i! 


(i 


: '  HI 


1- 


i     ! 
! 


il 


II 


162 


THE  DIFFERENTIAL  CALCULUS. 


Fio.  82. 


96.  Singular  Points  of  Curves,  If  we  conceiye  an  infini- 
tesimal circle  to  be  drawn  round  any 
point  of  a  curve  as  a  centre,  then,  in 
general,  the  curve  will  cut  the  circle  in 
two  opposite  points  only,  which  will 
be  180"  apart. 

But  special  points  may  sometimes  bo 
found  on  a  curve  where  the  infinitesimal  circle  will  be  cut  in 
some  other  way  than  this:  perhaps  in  more  or  less  than  two 
points;  perhaps  in  points  not  180°  apart.  These  are  called 
singular  points. 

The  principal  singular  points  are  the  following: 

Double-points;  at  which  a 
curve  intersects  itself.  Here  the 
curve  cuts  the  infinitesimal  circle 
in  four  points  (Fig.  33). 

Cusps;  where  two  branches  of 
a  curve  terminate  by  touching 
each  other  (Fig.  34).  Here  the 
infinitesimal  circle  is  cut  in  two  coincident  points. 

Stopping  Points;  where  a  curve  suddenly 
ends.     Here  the  infinitesimal  circle  is  cut  in 

,  .      ,  .    ,  Fio.  36. 

only  a  smgle  point. 

Isolated  Points;  from  which  no  curve  proceeds,  so  l._^ 
that  the  infinitesimal  circle  is  not  cut  at  all.  fio.  36. 

Salient  Points;  from  which  proceed  two  branches  making 
with  each  other  an  angle  which  is  neither  zero  nor  180". 
Here  the  infinitesimal  circle  is  cut  in  two  points  which  are 
neither  apposite  nor  coincident. 

There  may  also  be  multiple-points,  through  which  the  curve 
passes  any  number  of  times.  A  double-point  is  a  special  kind 
of  multiple-point. 

A  multiple-point  through  which  the  curve  passes  three 
times  is  called  a  triple-point. 


Fio.  83. 


Fio.  84. 


I  infini- 


)  cut  in 
lan  two 
3  called 


to.  84. 
35. 

o 

Fio.  36. 

making 
r  180". 
ich  are 

le  curve 
al  kind 

3  three 


ASYMPTOTES  AND  SINGULAR  POINTS. 


163 


97.  Condition  of  Singular  Points.  Let  {x^,  y,)  be  any 
point  on  a  curvo>  and  let  it  be  required  to  invutstigate  the 
question  whether  this  point  is  a  singular  one.  Wo  Urst  trans- 
form the  equation  of  the 
curve  to  one  in  polar  co- 
ordinates having  the  point 

(^0*  .*/o)  ^  ^^®  polo.  To  do 
this  we  put,  in  the  equation 
of  the  curve, 

x  =  x^-i-  pcoBd;)  jjj 
y  =  y,+  pemO.) 

The    resulting    equation 
between  p  and  6  will  be  the  fio.  87. 

equation  of  the  curve  referred  to  {x^,  y^)  as  the  pole.  More- 
over, if  we  assign  to  /o  a  fixed  value,  the  corresponding  value 
of  6  derived  from  the  equation  will  be  the  angle  6  showing 
the  direction  QP  from  Q  to  the  point  P,  where  the  circle  of 
radius  p  cuts  the  curve.  The  limit  which  6  approaches  as  p 
becomes  infinitesimal  will  determine  the  points  of  intersection 
of  the  infinitesimal  circle  with  the  curve. 

If,  now,  the  given  equation  of  the  curve  is 

F(x,  y)  =  0, 

then,  by  the  substitution  (1),  the  polar  equation  will  be 

F{x,  +  P  cos  e,y,  +  p  sin  6)  =  0.  (2) 

Now,  let  us  develop  this  expression  in  powers  of  p  by  Mac- 
laurin .  theorem.  Since  p  enters  into  (2)  only  through  x  and 
y  in  (1),  we  have 

dF      dF  dx    ,   dF  dy  JF  ,     .    JF_  „, 

-7-=-^--T-  +  -j--^=cos  6-^  +  sin  0-j~  =  F', 
dp       dx  dp      dy    dp  dx  dy 


(because  -^-  =  cos  6  and  -:-  =  sin  6], 
\  dp  dp  i 

Then 


il 


M 


:,"f 


I  ,i 


164 


THE  DIFFERENTIAL  CALCULUS. 


dp^  ~  dp  ™  Kdx"    dp      dxdy  dpi 

4_  sin  e{~   ^4.t.^  ^\ 
[dxdy  dp       dy^   dp) 

=  cos'  e^-  +  3  sin  19  cos  6^^-  +  sin'  ^^f  =  F". 
dx  dxdy  dy 

Noting  that  when  p  =  0  then  x  =  x^,  we  see  that  the  de- 
velopment by  Maclaurin's  theorem  will  be 

F[x,  y)  =  F{x^,  y,)  +  p(cos  e'-^  +  sin  ^^  j 

.   1    ,/     ,  J'F  .  „  .    1        .  d'F     ,     .  ,  J'F 

-\-  -p  \  cos  ff^—-,  +  2  Sin  6/  cos  ff-^ — -. h  sm  6-j—r 

2"^  \         dx/  dx^dy,  dy/ 

-\-  etc.  =  0. 

dW  dF 

Here  -j-  means  the  value  of  y-  when  x^  is  put  for  x,  etc. 

Because  (a-^,  yj  is  by  hypothesis  a  point  on  the  curve,  we 
have  F{x^y  ?/„)  =  0,  and  the  only  terms  of  the  second  member 
are  those  in  p,  p',  etc.  Thus  the  polar  equation  (2)  of  the 
curve  may  be  written 


FJP  +  i^o'V  +  Frp'  +  etc.  =  0,  ) 

5.  =  0.  f 


(3) 


li  ! 


OP  F:    +  FJ'p  +  F/'^p"  +  etc. 

To  find  the  points  in  which  the  curve  cuts  a  circle  of  radius 
p,  we  have  to  determine  ^  as  a  function  of  p  from  this  equa- 
tion. When  p  is  an  infinitesimal,  all  the  terms  after  the  first 
will  be  infinitesimals.  Hence,  at  tlie  limit,  ivhere  p  becomes 
infinitesimal  6  must  satisfy  the  equation 

f: = 0, 

dF 

fix 
which  ffives  tan  6  =  —  -~. 

dF 

Wo 

This  is  the  known  equation  for  the  slope  of  the  tangrnt  at 
{x^f  yj,  and  gives  only  the  evident  result  that  ia  general  the 


M 


ASYMPTOTES  AND  SINGULAR  POINTS. 


165 


curve  cuts  the  infinitesimal  circle  along  the  line  tangent  to 
the  curve  at  Q. 
But,  if  possible,  lot  the  point  (.^o^o)  be  so  taken  that 


(i) 


Then  we  shall  have  F/  ~  0,  and  the  equation  (3)  of  the 
curve  will  reduce  to 

F/'p  +  F/"p'  +  etc.  =  0, 
or  F/'    +  F/''p  +  etc.  =  0. 

Again^  letting  p  become  infinitesimal,  we  shall  have  at  the 
limit 


5T  //    


(VF  (PF 

cop'  (^~j—i  -\-  2  sin  0  cos  6 


fPF 


,     ,     -fsin'^,--=0. 


(5) 


Dividing  throughout  by  cos'  0,  we  shall  have  a  quadratic 
equation  in  tan  0,  which  will  have  two  roots.  8inco  each 
value  of  tan  0  gives  a  pair  of  opj)osite  points  in  wliich  the 
curve  may  cut  the  infinitesimal  circle,  and  since  (5)  depends 
on  (4),  we  conclude: 

The  necessary  condition  of  a  doiihle-jwi] it  is  that  the  three 
equations 

0, 


j^,,y)  =  0,    ^^^^=0,    ^3^'^) 


dx,  '  dij 

shall  he  satisfied  hy  a  sinyJe  pair  of  values  of  x  and  y. 

If  the  two  values  of  tan  6  derived  from  FJ'  =  0  are  equal, 
we  shall  have  either  a  cusp,  or  a  point  in  wliich  two  branches 
of  the  curve  touch  each  other.  If  the  roots  are  imaginary, 
the  singular  point  will  be  an  isolated  point. 

1)8.  Examples  of  Douhh'-poials.  A  curve  whose  equation 
contains  no  terms  of  less  tlum  the  second  degree  in  x  and  y 
has  a  singular  point  at  tlio  origin.  For  example,  if  the  equa- 
tion be  of  the  form 

F{x,  y)  ^-  Px'  +  Qxy  +  Bf  =  0, 
then  this  expression  and  its  derivatives  with  respect  to  x  and 
y  will  vanish  for  x  =  0  and  y  —  0. 


'r 


in 


;H 


Hi' 


I 


m 


»• 


f 


■  1  i 


i 


m 


?yk 


i) 


166 


THE  DIFFERENTIAL  CALCULUS. 


Let  us  now  investigate  the  double-points  of  the  curve 

(y'  -  a'Y  -  3aV  -  2ax'  =  0. 
We  have 


(1) 


dF 

dx 

dF 
dy 


Q{a^x  -j-  ax*)  =  —  6ax{a  -f  x); 

My*  -  «')  =  -^yiy  +  «)  (2/  -  «)• 


(3) 


The  first  of  these  derivatives  vanishes  for    x  =  0  or  —  a; 
The  second  of  these  derivatives  vanishes  for  y  =0,  —  aor-{-a. 

Of  these  values  the  original  equation  is  satisfied  by  the  fol- 
lowing pairs: 


^0    = 


0;         0;     -a;) 
a;     -{-a;         0; ) 


^0  =  -  «; 

which  are  therefore  the  co-ordinates  of  singular  points. 


(3) 


Fio.  38. 

Differentiating  again,  we  have 

d'F 


d'F 

ax 


^^"^^  ^=^'  s^ 


-  =  12?/'  -  4a'. 


Forming  the  equation  F"  =  0,  it  gives 


ASYMPTOTES  AND  SINGULAR  POINTS. 


167 


(12^'  -  4a')  tan'  6  =  6a'  -\-  12ax. 

Substituting  the  pairs  of  co-ordinates  (3),  we  find: 

At  the  point  (0,  —  or),  tan  6^  =  ±  ^  Vd; 

At  the  point  (0,  +  a),  tan  <9  =  ±  ^  V3; 

At  the  point  ( —  a,  0),  tan  0  —  ±     Vf. 

The  values  of  tan  0  being  all  real  and  unequal,  all  of  these 
points  are  double-points.     The  curve  is  shown  in  the  figure. 

Remark.  In  the  preceding  theory  of  singular  points  it  is 
assumed  that  the  expression  (2),  §  97,  can  be  developed  in 
powers  of  p.  If  the  function  F  is  such  that  this  development 
is  impossible  for  certain  values  of  x^  and  y^,  this  impossibility 
may  indicate  a  singular  point  at  (x^,  y^. 

99.  Curve-tracing.  We  have  given  rough  figures  of  va- 
rious curves  in  the  preceding  theory,  and  it  is  desirable  that 
the  student  should  know  how  to  trace  curves  when  their 
equations  are  given.  The  most  elementary  method  is  that  of 
solving  the  equation  for  one  co-ordinate,  and  then  substitut- 
ing various  assumed  values  of  the  other  co-ordinate  in  the 
solution,  thus  fixing  various  points  of  the  curve.  But  un- 
less the  solution  can  be  found  by  an  equation  of  the  first  or 
second  degree,  this  method  will  be  tedious  or  impracticable. 
It  may,  howevor,  commonly  be  simplified. 

1.  If  the  equation  has  no  constant  term,  we  may  sometimes 
find  the  intersections  of  the  curve  with  a  number  of  lines 
through  the  origin.     To  do  this  we  put 

y  =  mx 

in  the  equation,  and  then  solve  for  x.  The  resulting  values 
of  x  as  a  function  of  m  are  the  abscissas  of  the  points  in  which 
the  curve  cuts  the  line 

y  —  mx  =■  0. 
Then,  by  putting 

m—  ±1,     ±2,     etc.;    m  =  ±  i,     ±  |,     etc., 
we  find  as  many  points  of  intersection  as  we  please. 


it^ 


f 


I' 


9  i 


168 


THE  DIFFERENTIAL  GALCULU 


To  makt  this  method  practicable,  the  equations  which  tto 
have  to  solve  should  not  be  of  a  degree  higher  than  the  secoTjd. 

If  the  curve  has  a  double-point,  it  may  be  convenient  to 
take  this  point  as  the  origin. 

2.  If  the  equation  is  symmetrical  in  x  and  y  or  x  and  —  y, 
the  curve  will  be  symmetrical  with  respect  to  one  of  the  lines 
X  —  y  =  0  and  x  -\-  y  =  0. 

The  equation  may  then  be  simplified  by  referring  it  to 
new  axes  making  an  angle  of  45"  with  the  original  ones. 

The  equations  for  transforming  to  such  axes  are 

X  =  [x'  -\-  y')  sin  45°; 
y=(^'-  y')  «in  45°. 
Ajmlication  to  the  Folium  of  Descartes,     If,  in  the  equa- 
tion of  this  curve, 

x'  +  y'  =  3«a;y, 

we  put  y  =  mx,  we  shall  find 

oam  Sain^ 


X  = 


y 


We  also  find,  from  the  equation  of  the  curve  and  the  pre- 
ceding expressions  for  x  and  y  in  terms  of  w, 

dy  _  x^  —  a','      2m  —  m* 


dx       ax 

■~  y 

l-2m='* 

Then,  for 

771  = 

1, 

3 

x  =--  ^a; 

y 

3 

dy 
dx 

=  — 

•1. 

m  — 

o 

2 

x  =  ~a; 

y- 

4 

=  3^; 

dy 
dx 

= 

4 
5' 

m  — 

3 
2' 

3G 

^  =  35"> 

y 

54 

dy 
dx 

= 

33 

92* 

m=  — 

2, 

6 

X  =  -^a; 

y 

12 

dy 

dx 

=::  — 

20 
17- 

etc. 

etc. 

etc. 

etc. 

Thus  we  have,  not  only  the  points  of  the  curve,  but  the 
rt:.i»<.;er.ts  of  the  angle  of  direction  of  the  curve  at  each  point, 
which  will  assist  us  in  tracing  it. 


Bli 


'^  f  r 


'':Wlr^ 


THEORY  OF  JENVELOPFS. 


169 


seconu. 
lent  to 

id  -y, 
lie  lines 

g  it  to 

3S. 


le  equa- 


tho  pre- 


3 
2* 

0 

7* 


but  the 


I 
I 


i 


CHAPTER  XIV. 

THEORY  OF  ENVELOPES. 

100.  The  equation  of  a  curve  generally  contains  one  or 
more  constants,  sometimes  called  parameters.  For  example, 
the  equation  of  a  circle, 

{X  -  ay  ^{y-  by  =  r\ 

contains  three  parameters,  a,  h  and  r. 

As  another  example,  we  know  that  the  equation  of  a 
straight  line  contains  two  independent  parameters. 

Conceive  now  that  the  equation  of  any  line,  straight  or 
curve,  (which  we  shall  call  ^the  line"  simply,)  to  be  written 
in  the  implicit  form 

0(a;,  y,  a)  =  0,  (1) 

a  being  a  parameter.  By  assigning  to  a  the  several  values 
a,  a*,  a",  etc.,  we  shall  have  an  equal  number  of  lines  whose 
equations  will  be 

^{x,  y,  a)  =  0;     cf)(x,  y,  a')  =  0;     (f){x,  y,  a")  =  0;     etc. 

The  collection  of  lines  that  cr.n  thus  be  formed  by  assign- 
ing all  values  to  a  parameter  is  called  a  family  of  lines. 

Any  two  lines  of  the  family,  e.g.,  those  wliich  have  a  and 
Of'  as  parameters,  will  in  general  have  one  or  more  points  of 
intersection,  determined  by  solving  the  corresponding  equa- 
tions for  X  and  y.  The  co-ordinates,  x  and  y,  of  the  point  of 
intersection  will  then  come  out  as  functions  of  a  and  ex'. 

Su^^p<iF-e  the  two  parameters  to  approach  infmltesimally  near 
eech  oUu'iV  The  point  of  intersection  will  then  appi'oacli  a 
cei\aii)  limit,  which  we  investigate  as  follows; 


Ml 


t-" 


m 


■,'i 


i'''ti 


'Ii 


170 


* 


THE  DIFFERENTIAL  CALCULUS. 


Let  us  put 

a'  ■=  a  -\-  Aa, 

The  equations  of  the  lines  will  then  be 

<f>{Xy  t/,  a)  =  0    and     (p{x,  y,  a  -{■  /ia)  =0. 

If  we  develop  the  left-hand  member  of  the  second  equation 
in  powers  of  J  a  by  Taylor's  theorem,  it  will  become 

0(^.  y.  «•)  +  a;i^«  +  ji-.  1-3  +  etc.  =  0. 

Subtracting  the  first  equation,  dividing  the  remainder  by 
J  a,  and  passing  to  the  limit,  we  find 

d<p{x,  y,  a)  __  ^ 
da 

Hence  ^-he  limit  toward  which  the  point  (x,  y)  of  intersec- 
tion of  two  lines  of  a  family  approaches  as  the  difference  of 
the  parameters  becomes  infinitesimal  is  found  by  determining 
X  and  y  from  the  equations 


<t>(x,y,c)  =  0    and    M^|^)  =  0. 


(2) 


The  vail  es  of  x  and  y  thus  determined  will,  in  general,  be 
functions  of  a;  that  is,  we  shall  have 

^  =/,(«);  2/ =/,(«);  (3) 

which  will  give  the  values  of  the  co-ordinates  x  and  y  of  the 
iiLiitin;.{  p(.int  of  intersection  for  each  value  of  a. 

Now,  suppose  a  to  vary.  Then  x  and  y  in  (3)  will  also 
vary,  and  will  determine  a  curve  as  the  locus  of  x  and  y. 

Such  a  curve  is  called  the  envelope  of  the  family  of 
lines,  <p{xj  y,  a)  =  0. 

In  (3)  the  equations  of  the  curve  are  in  the  form  of  (2), 
§  76,  a  beii  g  the  auxiliary  variable.  By  eliminating  a  either 
from  (3)  or  (3),  we  hsve  an  equation  between  x  and  y  which 
will  be  the  equation  of  the  curve  in  the  usual  form. 


■T: 


THEORY  OF  ENVELOPES. 


171 


10 !•  Theorem.  The  envelope  and  all  the  lines  of  the 
family  which  generate  it  are  tangent  to  each  other. 

Geometrically  the  truth  of  this  will  be  seen  by  drawing  a 
series  of  lines  varying  their  position  according  to  any  con- 
tinuous law,  as  in  the  first  example  of  the  following  sec- 
tion. Taking  three  consecutive  lines  and  numbering  them 
(1),  (2)  and  (3),  it  will  be  seen  that  as  (1)  and  (3)  approach 
(;i)  their  points  of  intersection  with  (2)  approach  infinitely 
near  each  other.  Since  these  infinitely  near  points  of  inter- 
section also  belong  to  the  envelope,  the  line  (2)  passes  through 
two  infinitely  near  points  of  the  envelope  and  is  therefore  a 
tangent  to  the  envelope. 

Analytic  Proof.  The  equation  of  the  envelope  is  found  by 
eliminating  a  from  the  equations  (2),  and  we  may  conceive 
this  elimination  to  be  effected  by  finding  the  value  of  a  from 
the  second  of  these  equations  (2),  and  substituting  it  in  the 
first  equation.     That  is,  the  equation 

(f){Xy  y,  a)  =  0  (4) 

represents  any  line  of  the  original  family  when  we  regard  a 
as  a  constant;  and  it  represents  the  envelope  when  we  regard 
a  as  a  function  of  x  and  y,  satisfying  the  equation 


d(f){x,  y,  a)  _^ 
da 


(6) 


Let  the  value  of  a  derived  from  this  last  equation  be 

a  =  F{x,  y).  (6) 

Now,  to  find  the  slope  of  the  tangent  to  the  original  line  of 
the  family  at  the  point  {x,  y),  we  differentiate  (4),  regarding 
a  as  a  constant.     Thus  we  have 


d0,d^dy___^ 
dx       dy  dx 


or    It^^P^, 

dx  l)y(p' 


(7) 


If  the  original  line  is  a  straight  one,  this  equation  will  give 
its  slope. 
To  find  the  slope  of  the  tangent  to  the  envelope  at  the  same 


!     * 


:  \ 


m 


It  t 

I?  f 


f 


172 


THE  DIFFERENTIAL  CALCULUS. 


point,  we  differentiate  this  same  equation,  regarding  a  as  hav- 
ing the  value  (6).     Thus  we  have 


dcf)      d<p  dy       d(f)fda       da  dp  \  _  /^ 
dx       dy  dx       da\dx        dy    dx )  ^ 

d(t) 


(8) 


But,  because  --  =  0,  this  equation  will  also  give  the  value 


a  a 


(7)  for  the  slope;  whence  the  curves  have  the  same  tangent  at 
the  point  {x,  y),  and  so  are  tangent  to  each  other  at  this  point. 

103.  We  shall  now  illustrate  this  theory  by  some  examples. 

1.  I'd  find  the  envelope  of  a  straight  line  which  moves  so  that 
the  area  of  the  trianyle  lohich  it  forms  with  the  axes  of  co- 
ordinates is  a  constant. 


(()(X,  y,  «)  :=  -  -f.  |- 


1  =  0. 


(1) 


FiQ.  39. 

Since  the  area  of  the  triangle  is  half  the  product  of  the     ( 
intercepts  of  the  axes  cut  off  by  the  line,  this  product  is  also 
constant. 

Calling  a  and  b  the  intercepts,  the  equation  of  the  line  may 
be  written  in  the  form 


'.if 


THEORY  OF  ENVELOPES. 


173 


the  line  may 


Here  we  have  two  varying  parameters,  a  and  b,  while,  to 
have  an  envelope,  the  change  of  the  parameters  must  depend 
on  a  single  varying  quantity.  But  the  condition  that  the 
product  of  the  intercepts  shall  be  constant  enables  us  to  elimi- 
nate one  of  the  parameters,  say  b.    We  have,  by  this  condition, 


6  =  -, 

a 


(2) 


whence 


da 


c 


Now  differentiating  the  equation  (1)  with  respect  to  a,  re- 
garding b  as  a  function  of  a,  we  have 

(70  _       ^       y  db  _  cy  —  Wx  ^  y       ^  ^  a 
da  ~       rt"       b^  da  ~"      a^b^      ~  c        a"  ~    * 


(3) 


We  have  now  to  eliminate  a  from  the  equations  (1)  and  (3), 
using  (2)  to  eliminate  b  from  (1).  The  easiest  way  to  effect 
this  elimination  is  as  follows: 

From  (3)  we  have 

a'y  =  cx;    a  =  A/y  (4) 

Multiplying  (1)  by  a,  and  substituting  for  b  its  value  from 
(2),  we  have 

x4-—^  =  a, 
c 

Substituting  from  (4),  this  equation  becomes 

and  thus  the  equation  of  the  envelope  becomes 

xy  =  ^c, 

which  is  thai  of  an  hyperbola  referred  to  its  asymptotes. 

This  result  coincides  with  one  already  found  in  Analytic 
Geometry,  that  tangents  to  an  hyperbola  cut  off  from  the 
asymptotes  intercepts  whose  product  is  a  constant. 


^iil 


M 


174 


THE  DIFFERENTIAL  CALCULUS. 


M 


;i 


5f  1 


At  i      i 


2.   To  find  the  envelope  of  the  line  for  which  the  sum  of  the 
intercepts  cut  off  from  the  co-ordinate  axes  is  a  constant. 


X    > 


Fro.  40. 


Let  c  be  the  constant  sum  of  the  intercepts.  Then,  if  a  be 
the  one  intercept,  the  other  will  be  a  —  a.  Thus  the  equa- 
tion of  the  line  is 


-+- 

a      c 


y 


a 


=  1, 


in  which  a  is  the  varying  parameter. 
Clearing  of  fractions,  we  may  write  the  equation 
cf){x,  ijy  a)  =  cx-\-a{ij  —  X  —  r)  +  a'  =  0, 
d(f) 


whence 


da 


=  y  —  X  —  c  -\-  2it  =  0. 


From  the  last  equation  we  have 

a=:^x-y-{■  c); 
this  value  of  a  being  substituted  in  the  other  gives 

ex  —  'l{x  —  y  -\-  cY  —  0, 
or  {x  -  yY  -  M^  -\-  y)  +  c'  =  0. 


5  sum  of  the 
istant. 


'hen,  if  a  be 
s  the  equa- 


0, 


THEORY  OF  ENVELOPES. 


176 


This  equation,  being  of  the  second  degree  in  the  co-ordi- 
nates, is  a  conic  section. 
The  terms  of  the  second  degree  forming  a  perfect  square, 

it  is  a  parabola. 
The  equation  of  the  axis  of  the  parabola  is 

X  —  y  =  0. 

To  find  the  two  points  in  which  the  parabola  cuts  the  axis 
of  X  we  put  y  =  0,  and  find  the  corresponding  values  of  x. 
The  resulting  equation  is 

x'  -  2cx  -\-c^  =  0. 

This  is  an  equation  with  two  equal  roots,  x  =  c,  showing 
that  the  parabola  touches  the  axis  of  X  at  the  point  {c,  0). 
It  is  shown  in  the  same  way  that  the  axis  of  Y  is  tangent  to 
the  parabola. 

It  may  also  be  shown  that  the  directrix  and  axis  of  the 
parabola  each  pass  through  the  origin,  and  that  the  parame- 
ter is  V  2c. 

3.  If  the  difference  of  the  intercepts  cut  off  by  a  line  from 
the  axes  is  constant,  it  may  be  shown  by  a  similar  process 
that  the  envelope  is  still  a  parabola.  This  is  left  as  an  exer- 
cise for  the  student,  who  should  be  able  to  demonstrate  the 
following  results : 

(a)  When  the  sum  of  the  intercepts  is  a  positive  constant, 
the  parabola  is  in  the  first  quadrant ;  when  a  negative  con- 
stant, the  parabola  is  in  the  third  quadrant. 

{/3)  When  the  difference,  a  —  b,  of  the  intercepts  is  a  posi- 
tive constant,  the  parabola  is  in  the  fourth  quadrant;  when  a 
negative  constant,  in  tlie  second. 

(y)  The  co-ordinate  axes  touch  the  parabola  at  the  ends  of 
the  parameter. 

In  each  case  the  parabola  touches  each  co-ordinate  axis  at 
a  point  determined  by  the  value  of  the  corresponding  inter- 
cept when  the  other  intercept  vanishes,  and  each  directrix 
intersects  the  origin  at  an  angle  of  45"  with  the  axis. 


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M'i 


176 


THE  DIFFERENTIAL  CALCULUS. 


4.  Next  take  the  case  in  which  the  sum  of  one  intercept 
and  a  certain  fraction  or  multiple  of  the  other  is  a  constant. 
Let  m  be  the  fraction  or  multiplier.    We  then  have 

i  ^  ma  =  c  =  a  constant. 

The  equation  of  the  line  then  becomes 


«       c  —  7na 


1  =  0. 


Proceeding  as  before,  we  find  the  equation  of  the  envelope 

to  be 

(mx  —  yY  —  2c(mx  +  y)  -|-  c'  =  0, 

which  is  still  the  equation  of  a  parabola. 

6.  To  find  the  envelope  of  a  line  which  cuts  off  intercepts 
subject  to  the  condition 


-■4--  =  l 


(a) 


m  and  n  being  constants. 

We  may  simplify  the  work  by  substituting  for  the  varying 
intercepts  a  and  b  the  single  variable  parameter  a  determined 
by  either  of  the  equations 

m  n 

sm  Of  =  — ;     cos  a-—, 
a  _      b 

The  equation  of  the  varying  Lbj  will  then  oecome 
d>(x,  v)  =  —  sin  or  +  —  cos  or  =  1. 

By  differentiating  with  respect  to  a,  we  have 

d<f)      X  y   .  - 

—i-=z—  COS  a  --  —  sm  or  =  0. 
da      m  n 


(1) 


(8) 


We  may  now  eliminate  a  by  simply  taking  the  sum  of  the 
squares  of  these  equations,  which  gives 

-^'  +  ^=1 

the  equation  of  an  ellipse  whose  semi-axes  are  m  and  n. 


THEORY  OF  ENVELOPES. 


177 


6.  To  find  the  envelope  of  a  cij-cle  of  constant  radius  whose 
centre  moves  on  a  fixed  circle. 

For  convenience  let  us  take  the  centre  of  the  fixed  circle  as 
the  origin,  and  put: 
a,  b,  =  the  co-ordinates  of  the  centre  of  the  moving  circle; 
c  =  its  radius; 

d  =  the  radius  of  the  fixed  circle. 
The  equation  of  the  moving  circle  now  becomes 

{x-aY  +  {y-by-c*  =  0.  (1) 

By  differentiation  with  respect  to  a. 

The  condition  that  (a,  b)  lies  on  the  fixed  circle  gives 

a*  +  b*  =  d*,  (3) 

,  db  a 

whence  j-  =  —  r» 

da  b 

Then,  by  substituting  this  value, 

ay  —  bx  =  0.  (3) 

We  have  now  to  eliminate  a  and  b  from  (1),  (2)  and  (3). 
Firstly,  from  (1)  and  (2),  we  find 

£»•  +  y*  -  2ax  -  2by  =  c'  -  d\  (1') 

From  (2)  and  (3)  we  find  the  following  expressions  for  a 

and  b: 

xd  ,  _        yd 


a  = 


I/*'  +  «/•' 


5  = 


i^a;»  +  y* 

By  substitution  in  (1'),  and  putting  for  brevity 

r'  =  a:'  +  2/% 

we  find  r'  ±  2rd  ■\- d""  —  c\ 

Hence  r'  =  a;'  -[-  y'  =  (c  ±  rf  )', 

the  equations  of  two  circles  around  the  origin  as  a  centre, 
with  radii  c  ■\-  d  and  c  —  d. 


it 


■} 


M   ,j 


Hi'.! 


iff 


,! 


178 


THE  DIFFERENTIAL  CALCULUS, 


7.  Find  the  envelope  of  a  family  of  ellipses  referred  to  their 
centre  and  axes,  the  product  of  whose  semi-axes  is  equal  to 
a  certain  constant,  o*, 

Ans,  The  equilateral  hyperbola  xy  =  ^c*. 

8.  To  find  the  envelope  of  a  family  of  straight  lines,  such 
that  the  product  of  their  distances  from  two  fixed  points  is  a 
constant. 

Let  {a,  0)  and  {—a,  0)  be  taken  as  the  two  fixed  points, 
and  let  c'  be  the  constant.    Also,  let 


xcoB  a  •{- y  sin  a  —  p  =  0 


(1) 


be  the  eqr  tion  of  any  one  of  the  lines  in  the  normal  form, 
p  and  a  being  the  varying  parameters. 

The  distances  of  the  line  from  the  points  {a,  0)  and  (—a, 
0)  are  respectively 

—  p-\-a  cos  a    and     —  j»  —  a  cos  a. 

Hence  we  have  the  condition 

jo'  —  a'  cos'  a  =  c'.  (2) 

Differentiating  (1),  regarding  jt?  as  a  function  of  a,  we  have 

—  XBin  a  4-  y  cos  a f-  =  0. 

^  da 


From  (2)  we  obtain 


dp  _ 
da  ~" 


a"  Bin  a  cos  a 
p  • 


We  thus  have  the  three  equations 


X  COB  a -\-  y  Bin  a  —  p, 

,    («) 

a*  sin  a  cos  a 
X  Bin  a  —  y  COB  a  = , 

{*) 

p*  =  c*  -{■  a*  cos'  a 
=  c'  4-  «'  —  «*  siii"  ^f 


{•) 


from  which  to  eliminate  p  and  a. 


THEORY  OF  ENVELOPES. 


179 


To  effect  the  elimination  of  a  and  p  we  find  the  yalnes  of 
X  and  y  from  (a)  and  (b)  by  taking 

(a)  X  COS  or  +  (b)  X  sin  a    and    (a)  X  sin  or  —  (J)  x  cos  a. 
We  thus  find^  by  the  aid  of  (c), 

px  =  p*  cos  a  +  a'  sin"  a  coaa; 
,  ,  ,     jvcos  a 


y  =  c' 


Hence 


sin  a 


c*  +  a' 


COB  dr 


y_  _  sin  flf 
c*^    p    * 

If  we  multiply  the  first  of  these  equations  by  x  and  the 
second  by  y  and  add^  then  we  have 

g* L  ^'  —  ^  cos  ^  4-  y  sin  «  __ 

c«  +  a«  +  ?  -         ]r'"      ~  • 

Hence  the  equation  of  the  enyelope  is 

c'  -f  «•  ^  e    ^' 

This  represents  an  ellipse  whose  foci  are  the  two  fixed 
points. 

This  interpretation^  howeyer,  presupposes  that  the  product 

c'  of  the  distances  of  the  line  from  th«3  two  points  is  positive; 

that  is,  that  the  points  are  on  the  same  side  of  the  enveloping 

line.    If  the  product  is  negatiye,  the  equation  of  the  envelope 

will  be 

^      -  y'  -  1 
o«  -  c"      c'  ~  ^' 

which  is  the  equation  of  an  hyperbola. 

These  results  give  the  theorem  of  Analytic  Geometry  that 
the  product  of  the  distances  of  a  tangent  from  the  foci  of  a 
conic  is  constant. 


i! 


m 


r. 


;% 


\      \\ 


■       i  ■ 

H 
U 


Hr 


l.:l  .  I 


i! 


180 


THE  DIFFERENTIAL  CALCULUS. 


CHAPTER  XV. 

OF  CURVATURE;    EVOLUTES  AND  INVOLUTES. 

103.  Position;  Direction;  Curvature,  The  position  of 
any  point  P  on  a  curve  is  fixed  by  the  yalues  of  the  co-ordi- 
nates, X  and  y,  of  P,    This  is  shown  in  Analytic  Geometry. 

If  we  have  given,  not  only  x  and  y,  but  the  value  of  -^  for 

the  point  P,  then  such  value  of  the  derivative  indicates  the 
direction  of  the  curve  at  the  point  P,  this  direction  being  the 
same  as  that  of  the  tangent  at  P.  \ 

The  curve  may  also  have  a  greater  or  less  degree  of  curva- 
ture at  P.  The  curvature  is  indicated  by  a  change  in  the  di- 
rection of  the  tangent,  that  is,  in  the  value  of  ~,  when  we 

pass  to  an  adjacent  point  P^    But  such  change  in  the  value 

dv 
of  -p-  when  we  vary  x  is  expressed  by  the  value  of  the  second 


derivative 


dx'' 


If  this  quantity  is  positive,  the  angle  which 


'^he  tangent  makes  with  the  axis  of  X  is  increasing  with  x  at 
the  point  P,  and  the  curve,  viewed  from  below,  is  convex. 

If  -t4  ^s  negative,  the  tangent  u  diminishing,  and  the 

curve,  SC3U  from  below,  is  concave. 

To  sum  up:  If  we  take  a  value  of  the  abscissa  x,  then  the 
corresponding  value  of 

y  gives  the  position  of  a  point  P  of  the  curve; 
—  gives  the  direction  of  the  curve  at  P; 

-=-^  depends  upon  the  ctirvature  of  the  curve  at  P. 


Il 


CURVATURE;  E VOLUTES  AND  INVOLUTES.       181 


104.  Contacts  of  Different  Ordera,  Let  two  different 
curves  be  given  by  their  respective  equations: 

y  =f{x)    and    y  =  <p{x). 

If  for  a  certain  value  of  x,  which  value  call  x^,  the  two 
values  of  y  are  equal,  the  two  curves  have  the  corresponding 
point  in  common;  that  is,  they  meet  at  the  point  {x^,  y). 

If  the  values  of  -^  are  also  equal  at  this  point,  it  shows 

that  the  curves  have  the  same  direction  at  the  point  of  meet- 
ing.    They  are  then  said  to  touch  each  other. 

(Pv 
If  the  values  of  -^  are  also  equal  at  this  point,  the  two 

curves  have  also  the  same  curvature  at  this  point. 

To  show  the  result  of  these  several  equalities,  let  us  give 
the  abscissa  x^  (which  we  still  take  the  same  for  both  curves), 
an  increment  Ti,  and  develop  the  two  values  of  y  in  powers  of 

h  by  Taylor's  theorem.     To  distinguish  the  values  of  y^  -r-t 

etc.,  which  belong  to  the  two  curves,  we  assign  to  one  the 
suffix  0,  and  to  the  other  the  suffix  1.  Then,  for  the  one 
curve. 


2'  =  -^«  +  (l)„^+©/2+ 


/^\A" 


and,  for  the  other. 


+ 


+  l^-J,^!  +  ^*^-' 


{d:^y\  ¥ 
.n\ 


+  l^«i  Zy  +  «*«• 


The  difference  between  the  values  of  y'  and  y  is  the  inter- 
cept, between  the  two  curves,  of  the  ordinate  at  the  point 
whose  abscissa  is  x^  -f  li.     Its  expression  is 

--.+[(i).-{s>4e).-{a]r.+- 

Now,  consider  the  case  in  which  the  curves  meet  at  the 
point  P,  whose  abscissa  is  x^.     Then 

y,  -  ^0  =  0, 


v\ 


li 


I 


li  I 


182 


THE  DIFFSBENTIAL  CALCULUS. 


and  the  intercept  of  the  ordinate  will  be 

[(f),-  (I-).]* + '«""" '"  *■'  «*"•' 

which,  when  h  becomes  infinitesimal,  is  an  infinitesimal  of 
the  first  order. 

« 

If  we  also  have 


ldy\  _  ldy\ 

\dxi~  xdx); 


the  ordinates  will  differ  only  by  a  quantity  containing  h*  as  a 
factor,  and  so  of  the  second  order.     Hence: 

Wfien  two  curves  are  tangent  to  each  other,  they  are  sepa- 
rated only  by  quantities  of  at  least  the  second  order  at  an  in- 
finitesimal  distance  from  the  point  oftangency. 

In  the  same  way  it  is  shown  that  if  the  second  differential 
coefficient  also  'vanishes,  the  separation  will  be  of  the  third 
order,  and  so  on. 

Def  When  two  curves  are  tangent  to  each  other,  if  the 
first  n  1  ^.'erential  coefficients  for  the  two  curves  are  equal  at 
the  p  >  of  tangency,  the  curves  are  said  to  have  contact 
of  the  nth  order. 

Hence  a  case  of  simple  tangency  is  a  contact  of  the  first 
order.  If  the  second  derivatives  are  also  equal,  the  contact 
is  of  the  second  order,  and  so  forth. 

105*  Theorem.  In  contacts  of  an  even  order  the  two 
curves  intersect  at  the  point  of  contact ;  in  those  of  an  odd 
order  they  do  not. 

For,  in  contact  of  the  ?ith  order,  the  first  term  of  y'  —  y 
(§  104)  which  does  not  vanish  contains  A^^*  as  a  factor. 

If  n  is  odd,  7i  +  1  is  even,  and  y'  —  y  has  the  same  alge- 
braic sign  whether  we  *^^ake  h  positively  or  negatively.  Hence 
the  curves  do  not  intersect. 

If  n  is  even,  n-\-l\B  odd,  and  the  values  oi  y'  —  y  havf» 
opposite  signs  on  the  two  sides  of  the  point  of  contact,  thus 
showing  that  the  curves  intersect. 


i)i  .J. 


CURVATURE;  EV0LUTB8  AND  INVOLUTES.      183 

106.  Eadius  of  Curifature.  The  curvature  at  anyiK>int 
is  measured  thus:  We  pass  froi;:i  the  point  P  to  a  point  P'  in- 
finitesimally  near  it.  The 
curyature  is  then  measured 
by  the  ratio  of  the  change 
in  the  direction  of  the  tan- 
gent (or  normal)  to  the 
distance  PP*»    Let  us  put 

a  7^  the  angle  which 
the  tangent  at  P  m^ikes 
with  the  axis  of  X. 

a  -\-  da  =  the  same  angle  for  the  tangent  at  P\ 
ds  E  the  infinitesimal  dibtance  PP\ 
Then,  by  definition. 

Curvature  =  -j-, 
ds 


Fio.  41. 


Now,  because 
we  have,  by  differentiation. 


tan  a  =  -^, 
dx 


Bed'  a  da  =  -v^^dx. 


sec'  Of  =  1  +  tan'  a  —  1  -{- 


Also, 

and  da  =  y  ll  ■}-  —-Adx, 

From  these  equations  we  readily  derive 

dx* 


Curvature  =  —r 
ds 


(^+gr 


Now,  draw  normals  to  the  curve  at  the  points  P  and  /", 
and  let  C  be  their  point  of  intersection.  Because  they  are 
perpendicular  to  the  tangents,  th^  angle  POP'  between  theni 
will  be  da,  and  if  we  put 

p  =  PO, 


nil 


!i 


'  t 
1 1 


1  I 
( 


if! 

i    i* 


■I 


livfM 


■'if.; 


184 


THE  DIFFERENTIAL  CALCULUS.     * 


we  shall  have 


ds 


Hence  p  —  —-^ 

da      curvature 


PP'  =zds  =  pda. 


dx* 


The  length  p  is  called  the  radius  of  curvature  at  the 
point  P,  and  O  is  called  the  centre  of  curvature. 

Corollary.     The  centre  of  curvature  for  any  point  of  a 
curve  is  the  intersection  of 
consecutive   normals    cut- 
ting   the  curve    infinitely 
near  that  point,  p7 


Fio.  48. 


107.  The  Osculating 
Circle,  If,  on  the  normal 
PC  io  any  curve  at  the 
point  P,  we  take  any  point 
as  the  centre  of  a  circle 

through  Pf  that  circle  will  be  tangent  to  the  curve  at  P\ 
that  is,  it  will,  in  general,  have  contact  of  the  first  order 
at  P.  But  there  is  one  such  circle  which  has  contact  of  a 
higher  order,  namely,  that  whose  centre  is  at  the  centre  of 
curvature.  Since  this  circle  will  have  the  same  curvature 
at  P  as  the  curve  itself  has,  it  will  have  contact  of  at  least 
the  second  order  at  P, 

This  proposition  is  rigorously  demonstrated  by  finding  that 
circle  which  shall  have  contact  of  the  second  order  with  the 
curve  at  the  point  P. 

Let  us  put 

Xf  y,  the  co-ordinates  of  P; 

jo  =  ~-  f or  the  curve  at  the  point  P; 

q  =  ^  for  the  curve  at  the  point  P. 


CURVATURE;  E VOLUTES  AND  INVOLUTES.       185 

These  last  two  quantities  are  found  by  differentiating  the 
equation  of  the  curre.  * 

Now,  ~  and  -i-^  must  have  these  same  vnlues  at  the  point 

{x,  y)  in  the  case  of  the  circle  having  contact  of  the  second 
order  (§  104). 
Let  the  equation  of  this  circle  be 

(X  -  aY  +  (y  -  hy  =  r\  (a) 

By  differentiation,  we  have 

(x  -  a)dx  +  (y  -  h)dy  =  0, 


whence 


dy  __  X  —  a  _ 
dx  ^  b  —  y 


=  p. 


w 


Differentiating  again, 

^  _  _i_  ,  (^  ~(i)dy  _  (y  -  by  H-  (r  -  a)* 
dx'-b-y'^  (b-yydx  -  {y-  by 

From  (b)  combined  with  (a)  we  find 

{x  -  ay  _       r* 


w 


1+y  =  l-f 


I  _ 


{y  -  by      (y  -  b) 


ty 


■■■   (!+/)•  = 


(y  -  *)■■ 

Dividing  this  by  (c)  gives 

^  _  a+py 

r  = , 

Q 

the  equivalent  of  the  expression  already  found  for  the  radius 

of  curvature. 

Hence  if  we  determine  a  circle  by  the  condition  that  it 

shall  have  contact  of  the  second  order  with  the  curve  at  the 

point  P,  its  radius  will  be  equal  to  the  radius  of  curvature. 

This  circle  is  called  the  osculating  circle  for  the  point  P. 

Each  point  of  a  curve  has  its  osculating  circle,  determined 

by  the  position,  direction  and  curvature  at  that  point. 


ill 


)   ,; 


fi 


'\  \ 


n 


M 


; 


lee 


THK  DTFFKRKNTIAL  CALCULUS. 


J   f 


Cor.  Tho  osculating  circle  will,  in  general,  intersect  the 
curve  at  the  point  of  contact,  for  it  has  contact  of  the  second 
ord3r. 

This  may  also  bo  seen  by  reflecting  that  tho  curvature  of  a 
curve  is,  in  general,  a  continuously  varying  quantity  as  we 
pass  along  the  curve,  and  that,  at  the  point  of  contact,  it  is 
equal  to  the  curvature  of  the  circle.  Hence,  on  one  side  of 
the  point  of  contact,  the  curvature  of  the  curve  is  less  than 
that  of  the  circle,  and  so  the  curve  passes  without  the  circle; 
and  on  the  other  side  the  curvature  of  the  curve  is  greater, 
and  thus  the  curve  passes  within  the  circle. 

If,  however,  the  curvature  should  be  a  maximum  or  a 
minimum  at  the  point  of  contact,  it  v/ill  either  increase  on 
both  sides  of  this  point  or  diminish  on  both  sides^  whence 
the  circle  will  not  intersect  the  curve,  \ 

108.  Radius  of  Curvature  when  the  Abscissa  is  not  taken 
as  the  Independent  Variable.  Suppose  that,  instead  of  x, 
some  other  variable,  ti,  is  regarded  as  the  independent  vari- 
able.   We  then  have 

Now,  it  has  been  shown  that,  in  this  case,  we  have  (§56) 


d^y  dx       d*x  dy 
d*y  _  du*  du      du^  du 

\dul 


(2) 


Also,  we  have 


fdy_Y       fdxV       /dyV 

1   _L  (^y\  —  1   _L  ^^"^^^  ^     _    \C?^<  /  \du  I  .    . 

"^  \d^}  ~    "^  idxv  ~        TdP\'      '        ^"^^ 

\du  I  \du  I 

These  expressions  being  substituted  in  the  expression  for 
the  radius  of  curvature,  it  becomes 


CURVATURE;  E VOLUTES  AND  INVOLUTES.       187 


*  )  i 


P  =  - 


\  \du  I  ^  [iiu  I  \ 
(Py  dx  d^x  dy 
du*  du       du'  du 


(4) 


lOO.  Radius  of  Curvature  of  a  Curve  referred  to  Polar 
Co-ordinates.  Let  the  equation  of  the  curve  be  given  in  the 
form 

The  preceding  expression  (4)  may  be  employed  in  this  case 
by  taking  the  angle  0  as  the  independent  variable.  By  differ- 
entiating the  expressions 

X  =  r  OS  9, 
y  —  r  sm  6, 

regarding  r  as  a  function  of  0,  we  find,  when  we  put,  for 
brevity, 


dr 


d'r 


—  —  r  sin  ^  -|-  r'  cos  6) 


dx 

Te 

II  =  (r"  -  r)  cos  e  -  2r'  sin  ff; 

-^  =  r  cos  6'  +  r'  sin  6*; 
at/ 

A  ^  ^^f,  _  ^.)  gijj  0  _^  2r'  cos  e. 

By  substituting  these  derivatives  with  respect  to  0  for  those 
with  respect  to  u  in  (4)  and  performing  easy  reductions,  we 
find 


-  _Jrl±i!!)L__ 

^-r'  -  rr"  +  2r"  ~ 


(5) 


which  is  the  required  expression  for  the  radius  of  curvature. 


'i 


'  M 


'P 


11 


:\ 


tl 


n 


188 


THE  DIFFERENTIAL  CALCULUS 


EXAMPLES    AND   EXERCISES. 

1.  The  Parabola.  To  find  the  radius  of  curvature  of  a 
curve  at  any  point,  we  have  to  form  the  value  of  p  from  the 
equation  of  the  curve.     The  equation  of  the  parabola  is 

''  -  2px, 


whence  we  find 


y 


dx       y ' 

cly_ 

dx^~ 


Then,  by  substituting  in  the  expression  for  /),  we  find 


n? 


^  _  (y'  +  f) 


\ 


the  negative  sign  being  omitted,  because  we  have  no  occasion 
to  apply  any  sign  to  p. 
At  the  vertex  y  =  0,  whence 

p=p. 

Hence,  at  the  vertex,  the  radiis  of  curvature  is  equal  to 
the  semi-parameter,  and  the  cent.'e  o!  cuivature  is  therefore 
twice  as  far  from  the  vertex  as  the  i'ocus  is. 

3.  Show  that  the  radius  of  curvature  at  any  point  {x,  y)  of 
an  ellipse  is 

^  ~  a*b' 

and  show  that  at  the  extremities  of  the  axes  it  is  a  third  pro- 
portional to  the  semi-axes. 

3.  Show  that  the  algebraic  expression  for  p  is  the  s^me  in 
the  case  of  the  hyperbola  as  in  that  of  the  ellip?3e. 

4.  What  must  be  the  eccentricity  of  an  ellipse  that  the  cen- 
tre of  curvature  for  a  point  at  one  end  of  the  minor  axis  may 
lie  on  the  other  end  of  that  axis?  Ans.  e  =  i^\. 


!.;!l 


)  occasion 


I  St^me  m 


CURVATURE;  EV0LUTE8  AND  INVOLUTES.       189 

5.  Show  that  in  the  case  supposed  in  the  last  problem  the 
radius  of  curvature  at  an  end  of  the  major  axis  will  be  one 
fourth  that  axis. 

6.  The  Cycloid.  By  differentiating  the  equations  (?),  §80, 
of  the  cycloid,  we  find 

dx 


-^—  =  a  —  a  cos  w  =  y, 
du  ^ 

d^x       dy 

du       du 

-~z  =  a  cos  ti. 
du 


(2) 


Then,  by  substituting  in  (4)  and  reducing,  we  find,  for  the 
radius  of  curvature, 

p  =  2^a  i^l  —  cos  w  =  4a  sin  ^u. 

We  see  that  at  the  cusp,  0,  of  the  cycloid,  where  u  =  0, 
the  radius  of  curvature  also  becomes  zero. 

7.  The  Archimedean  Spiral.     Show  from  (5)  that  the  ra- 
dius of  curvature  of  this  spiral  (r  =  ad)  is 

a(l  +  n^ 

8.  The  Logarithmic  Spiral,      The  equation  of  the  loga- 
rithmic spiral  being 

10 

r  =  ae  , 
show  that  the  radius  of  curvature  is 


p  =  r  VT+T. 

Hence  show  that  the  line  drawn  from  the  centre  of  curva- 
ture of  any  point  P  of  the  spiral  to  the  pole  is  j.>erpendicular 
to  the  radius  vector  of  the  point  P. 

9.  Show  that  the  radius  of  curvature  of  the  lemniscate  in 
terms  of  polar  co-ordinates  is 


P  = 


a 


«■ 


3  l^cos  36*      3r- 


! 


H. 


■ '  I 


'11  II 


t     I 


!-1 


'■  1 


tf 


!  ' 


4 1     'nm  4 

^  mi 


190 


THE  DIFFERENTIAL  CALCULUS. 


110.  Evolutes  and  Involutes.  For  every  point  of  a  curve 
there  is  a  centre  of  curvature,  found  by  the  preceding  for- 
mul8B.  The  locus  of  all  such 
centres  is  called  the  evolute 
of  the  curve. 

To  find  the  evolute  of  a 
curve,  let  (x^y^)  be  the  co-ordi- 
nates of  any  point  P  of  the 
curve  ;  PC,  the  radius  of  cur- 
vature for  this  point;  and  a, 
the  angle  which  the  tangent 
at  P  makes  with  the  axis  of  X, 
Then,  for  the  co-ordinates  of  fio.  43. 

C,  we  have  1 

a;  =  a;^  —  p  sin  a; 
y  =  yj  +  PCOB  a. 

Substituting  for  p  its  value  (§  106),  and  for  sin  a  and  cos  a 
their  values  fiom  the  equation 


we  find 


tan^  =  g. 


1  + 


^.y. 


dy; 

dx/ 


dx; 


1  + 


y  =  y,+ 


dyl 

dx; 


d% 
dx; 


(1) 


If  in  the  second  members  of  these  equations  we  substitute 
the  values  of  the  derivatives  obtained  from  the  equation  of 
the  curve,  we  shall  have  two  equations  between  the  four  vari- 
ables X,  y,  X,  and  y^.  By  eliminating  x^  and  y  from  these 
equations  and  that  of  the  given  curve,  we  shall  have  a  single 
equation  between  x  and  y,  which  will  be  that  of  the  evolute. 


;  X. 


CURVATURE;  EV0LUTE8  AND  INVOLUTES.       191 

111.  Cane  of  ail  Auxiliav^  Variable.     If  the  equation  of 
the  curve  is  expressed  by  an  auxiliary  variable,  we  have  to  make 

in  (1)  the  same  substitution  of  the  values  of  ~,  ~J,  etc. , 

as  in  §  108.     Thus  we  find,  instead  of  (1), 

(dxy 


du    du" 


x=  X, 


dy^     \du 


u I   ^  \du  I 


dud'y^dx^       d^x^dy/ 
du^  du       du''  du 


IdxV       iclyV 
_  dx^      \du  /         \dn  I 

^  ~  ^'       du  iv'y^  dx^  d'x^  dy^* 

dit*  du  du^  die 


m 


T 


which  are  the  equations  of  the  evolute  in  the  same  form. 


EXAMPLES  OF  EVOLUTES. 

112.  T7ie  Evolute  of  the  Parabola.  If  we  substitute  in 
(1)  for  the  derivatives  of  y^  with  respect  to  x^  the  values 
already  found  for  the  parabola,  these  equations  (1)  become 

,       ,   V/  ,  3  y; 

*     ^       I)      ^      ^  p 

We  now  have  to  eliminate  y^  from  these  two  equations,  x^ 
having  already  been  eliminated  by  the  equation  of  the  curve. 
They  give 

Equating  the  cube  of  the  first  equation  to  the  square  of  the 
second,  we  find,  for  the  equation  of  the  evolute  of  the  parabola, 

,_  8  {X  -  pY 


t  "1 


y  =27 


V 


W'^ 


''  1111 

If 

■m 


192 


THS!  DIFFERENTIAL  CALCULUS. 


113,  Evolute  of  the  Ellipse.    From  the  equation  of  the 
ellipse,  we  find 


a'y/    dx; 


.1    •• 


By  substituting  in  (1)  and  reducing,  we  find 

b*x;\  __     a*b'  -  a*y;  -  b,x/ 


X 


'       0   \      ci>  y/J 


a'y//         '  a*b^ 

Remarking  that  the  equation  of  the  ellipse  gives 

a*b'  -  a*y;  =  a'b'x% 
and  putting  6^  =  a^  —  b*, 

the  preceding  equation  becomes 


X  = 


a*  • 


In  the  same  way  we  get 

In  this  case  the  easiest  way  to  effect  the  elimination  of  x^ 
and  y,  is  to  obtain  the  values  of  these  quantities  from  (a) 
and  {b)f  and  then  substitute  them  in  the  equation  of  the 
ellipse.     From  (a)  and  (b),  we  find 

which  values  are  to  be 
substituted  in  the  equa- 
tion 

a'  +  J'  "  ■^• 

We  thus  find,  for  the 
equation  of  the  evolute  of 
the  ellipse. 

The  figure  shows  the 
form  of  the  curve.     The  following  T)roperties  should  be  de- 
duced by  the  student. 


CURVATURE;  EV0LUTE8  AND  INVOLUTES.       193 

(a)  The  evolute  lies  wholly  within  the  ellipse,  or  cuts  it  (as 
in  the  figure),  according  as  e'  <  i  or  e*  >  \. 

{b)  The  ratio  CD  :  AB  (which  lines  we  may  call  axes  of 
the  evolute)  is  the  inverse  of  the  ratio  of  the  corresponding 
axes  of  the  ellipse. 

114.  Evolute  of  the  Cycloid.  Here  we  have  to  apply  the 
formulae  (2)  for  the  case  of  a  separate  independent  variable. 
Substituting  in  (2)  the  values  of  the  derivatives  already  given 
for  the  cycloid,  we  shall  find 

d^y  dx  _  d^  ^  —  _    VI  _  ^• 

dii^  du       du*  du  ~         ^  '' 

X  =  x^  -[-  2a  sin  u  =  a(u  -f  sin  ti); 

y  =  y^  —  2a{l  —  cos  u)  =  —  a{l  —  cos  u). 

These  last  two  equations  are  those  of  the  evolute. 

Let  us  investigate  its  form.      For  «*  =  0  we  have  x  =  0 
and  y  =  0,  whence  the 
origin  is  a  point  of  the 
curve. 

For  u  =  Tt  we  have 

X  =  aTt; 
y=  -2a; 

giving  a  point  0,  below 
the  middle  of  the  base  of 
the  cycloid,   at  the  dis- 
tance 2a.     Let  us  take  this  point  as  a  new  origin,  and  call 
the  co-ordinates  referred  to  it  x'  and  y'.     We  then  have 

x'  ■='X  —  an  ■=  a(d  —  tt  -f  sin  6)\ 
y'  =  y  -|-  2a  =  a{l  +  cos  6). 


Fio.  45. 


If  we  now  put 
these  equations  become 


e'  =  e-7r, 


I 


•'1 1 


'i'i 

'   HI 


<  < 


lii'ni 


194 


THE  DIFFEBENTIAL  CALCULUS, 


X*  =  a{B'  -  sin  6/'); 

y'  =  «(i  -  cos  ey, 

which  are  the  equations  of  another  cycloid,  equal  to  the 
original  one,  and  similarly  situated.  The  cycloid  therefore 
posesses  the  remarkable  property  of  being  identical  in  form 
with  its  own  evolute. 


115.  Fundamental  Properties  of  the  Evolute. 

Theorem  I.     The  involute  of  a  curve  is  the  envelope  of  its 
normals. 

As  we  move  along  a  curve,  the  normal  will  be  a  straight 
line  moving  according  to  a  certain  law  depending  upon  the 
form  of  the  curve.  This  line  will,  in  general,  have  an  en- 
velope, which  envelope  will  be^,  '>y  definition,  the  locus  of  the 
point  of  intersection  of  consecutive  normals.  But  this  point 
has  been  shown  to  be  the  centre  of  curvature,  whose  locuc  is, 
by  definition,  the  evolute. 
Hence  follows  the  theorem. 

Corollary.  2'he  nor- 
mals io  a  curve  are  tan- 
gents to  its  evolute.  For 
this  has  been  shown  to  be 
true  of  a  moving  line  and 
its  envelope. 

Theorem  II.  Tf  the  os- 
culating circle  move  around 
the  curve,  the  motion  of  its 
centre  is  along  the  line  join- 
ing that  centre  to  the  ^oint 
of  contact. 

This  theorem  will  be 
made  evident  by  a  study 
of  the  figure.  If  the  line 
P^C^  be  one  of  the  nor- 
mals from  the  point  of  contact  P,  to  the  centre,  then,  since 


■  a.  46. 


CUli VA TUBE ;  F.VOL JITE8  AND  INVOL UTES.       1 95 


•  n 


[ual  to  the 
(1  therefore 
5al  in  form 


wlope  of  its 

a  straight 
'  upon  the 
ive  an  en- 
sous  of  the 
this  point 
0  locuc  is, 


an,  since 


this  normal  is  tangent  to  the  locus  of  the  centre,  it  will  bo  the 
line  abng  which  the  centre  is  moving  at  the  instant. 

TiiEOMEM  III.  The  arc  of  the  cvolute  contained  between 
any  two  points  is  equal  to  the  difference  of  the  radii  of  the 
osctdating  circles  whose  centres  are  at  these  points. 

For»  if  we  suppose  the  points  C„  C,,  etc.,  to  approach  in- 
finitesimally  near  each  other,  then,  sinro  the  infinitesimal 
arcs  Cf\,  ^-fi^y  ®^^'*  ^^'^  coincident  with  those  successive 
radii  of  the  osculating  circle  which  are  normal  to  the  curve, 
these  radii  are  continually  diminished  by  these  same  infini- 
tesimal amounts. 

The  analytic  proof  of  Theorems  II.  and  III.  is  as  follows: 
Let  the  equation  of  the  osculating  circle  be 

{x  -  «)«  +  (y  ^  lY  =  P\ 
where  a  and  h  are  the  co-ordinates  of  the  centre  of  curvature, 
and  therefore  of  a  point  of  the  e  •.  olute. 

The  complete  differential  of  this  equation  gives 

{x  —  a)  (dx  —  da)  +  (^  —  5)  {dy  —  db)  -•  pdp.  (a) 
If,  in  this  equation,  we  suppose  x  and  y  to  be  the  co-ordi- 
nates of  the  point  of  contact  of  the  circle  with  the  curve,  then 
dx  and  dy  will  have  the  same  value  at  this  point  whether  we 
conceive  them  to  belong  to  the  circle,  supposed  for  the  mo- 
ment to  be  fixed,  or  to  the  curve.    But  in  the  fixed  circle  we 

have 

{x  —  a)dx  +  (;/  —  b)dy  =  0.  (b) 

Subtracting  this  equation  from  {a)  and  dividing  by  p,  we  find 

y  -b 


c  —  a 
da  ■+■ 

P 


-db  =  —  J/o, 


(c) 


which  iS'  a  relation  between  the  differential  of  the  co-ordi- 
nates of  the  centre  and  the  differential  of  the  radius.  Now, 
if  we  put  /?  for  the  angle  which  the  normal  radius  makes 
with  the  axis  of  X,  we  have 


X 


a 


=  cos  /3', 


V  -~  ^ 


=  sin  y^. 


(d) 


1 


;i 


';  I 


t  ;i 


196 


THE  niFFEllENTIAL  CALCULUS. 


i: 


\\   I 


I 


I 


S     i 


But  this  same  normal  radius  is  a  tarigent  to  the  evolute. 
If  wo  call  (T  the  arc  of  tho  evolute,  wo  find  by  a  simple  con- 
struction da  =  cos  pd(T;  db  —  sin  {ido\ 

Multiplying  these  equations  by  cos  /?  and  sin  /',  respectively, 
and  adding,  we  find 

da  =  cos  ftda  -\-  sin  ^dh. 

Comparing  (c)  and  {d),  we  find 

d(T  :=  —  dp, 
or  d{(T  -\-  p)  =  0. 

Now,  a  quantity  whose  difierential  is  zero  is  a  constant. 
Hence  we  always  have 

a  -\-  p  =:  constant, 
or  (T  =  constant  —  p,  \ 

If  wo  represent  by  c,  and  a*,  the  arcs  from  any  arbitrary 
point  of  the  involute  to  the  two  chosen  points,  and  by  p,  and 
/o,  the  values  of  p  for  these  points,  we  have 

(T,  =  const.  —  p,; 
(T,  =  const.  —  Pj,. 

.•.    (T,  —  (T^  =  p,  -p,, 

or  the  intercepted  arc  equal  to  the  difference  of  the  radii,  as 
was  to  be  proved. 

It  must  be  remarked,  however,  that  whenever  we  pass  a 
cusp  on  the  evolute,  we  must  regard  the  arc  as  negative  on 
one  side  and  positive  on  the  other.  In  the  case  of  the  ellipse, 
for  example,  those  radii  will  be  equal  which  terminate  at 
equal  distances  on  the  two  sides  of  any  cusp,  as  ^,  ^,  C  or 
D,  and  the  intercepted  arc  must  then  be  taken  as  zero. 

116.  Invohites.  The  involute,  of  a  curve  0  is  that 
curve  which  has  C  as  its  evolute. 

The  fundamental  property  of  the  involute  is  this:  The 
involute  may  be  formed  from  tiie  evolute  by  rolling  a  tangent 


CUBVATURUE;    EV0LVTE8  AND  INVOLUTES.       197 

line  upon  the  latter.  A  point  P  on  the  rolling  tangent  will 
then  describe  the  involute. 

This  will  be  been  by  reference  to  Fig.  4G.  The  rolling  line, 
being  tangent  to  the  evolate,  coincides  with  the  radius  /*,6'„ 
and  as  it  rolls  along  the  evolute  into  successive  positions, 
/^,C„  P,C„  etc.,  the  motion  of  the  point  P  is  continually 
normal  to  its  direction. 

It  will  also  bo  seen  that  the  radius  of  curvature  of  the  in- 
volute at  each  point  is  equal  to  the  distance  PC  from  P  to 
the  point  of  contact  with  the  evolute. 

The  conception  may  be  made  clearer  by  conceiving  the 
rolling  line  to  be  represented  by  a  string  which  is  wrapped 
around  the  evolute.  The  involute  is  then  formed  by  the  mo- 
tion of  a  point  on  the  string. 

The  general  method  of  determining  the  involutes  of  given 
curves  involves  the  integral  calculus. 


:;1 


'•:   i 


i 


.lii 


PART  II. 


THE   INTEGRAL  CALCULUS. 


l:       |: 


■It 


■  I 


I 

ii 


li 


I 


Mi 


THE 

117. 

a  functi( 


we  may. 


which  w< 

In  the 

We  have 

and  the  j 
entiated, 
Every  i 
The  pr 

The  op 

called  "i: 

function 

pression 


means:  th 
F'{x)dx. 


'  t 


PART  II. 
THE  INTEGRAL  CALCULUS, 


w 


CHAPTER  I. 

THE  ELEMENTARY  FORMS  OF  INTEGRATION. 

117.  Definition  of  Integration,    Whenever  we  have  given 
a  function  of  a  variable  x^  say 

u  =  F(x), 

we  may,  by  differentiation,  obtain  another  function  of  a;, 

<^W  nil    \ 

n  =  -^  (^)' 

which  we  call  the  derived  function. 

In  the  integral  calculus  wo  consider  the  reverse  process. 
We  have  given  a  derived  function 

and  the  problem  is:  What  function  or  functions,  token  differ- 
entiated, will  have  F'(x)  as  their  derivative? 
Every  such  function  is  called  an  integral  of  F'(x), 
The  process  of  finding  the  integral  is  called  integration. 

The  operation  of  integration  is  indicated  by  the  sign   /  , 

called  "  integral  of,''  written  before  the  product  of  the  given 
function  by  the  differential  of  the  variable.  Thus  the  ex- 
pression 

fF'(x)dx 

means:  that  function  whose  differential  with  respect  to  x  is 
F'(x)dx. 


u 


I    1  ,i 


li 


'W 


M\\ 


202 


THE  INTEGRAL  CALCULUS. 


^    111 


i:H 


Calling  u  the  required  function,  then  if  *ye  have 


we  must  also  have 


^^^*  T7//      \ 


w 


As  examples: 
Because 


we  have 
Because 

we  have 


d{x^)  =  2xdx, 
'  2xdx  =  x^. 


d{ax^  -{-Ix-^-  c)  =  {2ax  -j-  b)dx, 

I  {2ax  +  ^)dx  =  ax*  -{-  bx  -{-  c. 

And,  in  general,  if,  by  differentiation,  we  have 

dF{z)  =  F'{x)dx,  \ 

we  shall  have  /  F'{x)dx  =  F{x), 

118.  Arbitrary  Constant  of  Integration,  The  following 
principle  is  a  fundamental  one  of  the  integral  calculus: 

If  F{x)  is  the  integral  of  any  derived  function  of  the  va- 
riable X,  then  every  function  of  the  form 

F{x)-\-h 
h  being  any  quantity  whatever  independent  of  Xy  will  also  be 
an  integral. 

This  follows  immediately  from  the  fact  that  h  will  dis- 
appear in  differentiation,  so  that  the  two  functions 

F(x)    and    F{x)^h 

have  the  same  derivative  (cf.  §  24). 

The  same  principle  may  be  seen  from  another  point  of 
view :  Since  the  problem  of  differentiation  is  to  find  a  func- 
tion which,  being  differentiated,  will  give  a  certain  result, 
and  since  any  quantity  independent  of  the  variable  which 
may  be  added  to  the  original  function  will  have  disappeared 
by  differentiation,  it  follows  that  wc  must,  to  have  the  most 


THE  ELEMENTARY  FORMS  OF  INTEGRATION.     203 


Hi 


general  expression  for  the  inte^al,  add  this  possible  but  un- 
known quantity  to  the  integral. 

The  quantity  thus  added  is  called  an  arbitrary  constant. 
But  it  must  be  well  understood  that  the  word  constant  merely 
means  independent  of  the  variable  with  reference  to  which 
the  integration  is  performed. 

It  follows  from  all  this  that  the  integral  can  never  be  com- 
pletely found  from  the  differential  equation  alone,  but  that 
some  other  datum  is  needed  to  determine  the  arbitrary  con- 
stant and  thus  to  complete  the  solution. 

Such  a  datum  is  the  value  of  the  integral  for  some  one 
value  of  the  variable.  Let  F{x)  -\-  h  be  the  integral,  and  ist 
it  be  given  that 

when  X  —  a,    then    the  integral  =  K, 

We  must  have,  by  this  datum, 

F{a)  -{-h  =  K, 
which  gives  h  =  K  —  F{a), 

and  thus  determines  h. 

Remark.  Any  symbol  may  be  taken  to  represent  the  ar- 
bitrary constant.  The  letters  c  and  h  are  those  most  gener- 
ally used.  We  may  affix  to  it  either  the  positive  or  the  nega 
tive  sign,  and  may  represent  it  by  any  function  of  arbitrary 
but  constant  quantities  which  we  find  it  convenient  to  intro- 
duce. It  is  often  advantageous  to  write  it  as  a  quantity  of  the 
same  kind  as  the  variable  which  is  integrated. 

119.  Inter/ration  of  Entire  Fxmctions, 
Theorem  I.     The  integral  of  any  j)ower  of  a  variable  is 
the  power  higher  by  unity,  divided  by  the  increased  exponent. 
In  symbolic  language,  we  have 

x^dx  =       ,   ^  -f-  h. 


f 


n-\-l 


X 


n  +  l 


«' 


For,  by  differentiating  the  expression 


-|-  h,  we  have 


;  1 


il 


IIP, 


204 


THE  INTEGRAL  CALCULUS. 


I;.  I 


i\ 


Theorem  II.     Any  constant  factor  of  the  given  differen- 
tial may  be  written  before  the  sign  of  integration. 
In  symbolic  language. 


faF\x)dx  =  aCF\x)dx, 


This  is  the  converse  of  the  Theorem  of  §  23.  By  that 
theorem  we  have 

d{aF{xy)  =  adF{x), 

from  which  the  above  converse  theorem  at  once  follows. 
In  the  special  case  «  =  —  1  we  have 

J-  F'{x)dx  =  J*F{x)d(-  x)  =  -  J*F'{x)dx. 

Hence  the  corollary:  If  the  integral  is  preceded  by  the  nega- 
tive sign  we  may  place  that  sign  before  either  the  derived 
function  or  the  differential. 

Theorem  III.  If  the  derived  function  is  a  sum  of  several 
terms,  the  integral  is  the  sum  of  the  separate  integrals  of  the 
terms. 

In  symbolic  language, 

/*(jr-{    r+  Z-\-  .  .  ,)dx  -  f  Xdx-\- J Ydx^  C Zdx^  . . 

This,  again,  is  the  converse  of  Theorem  II  of  §  22. 

The  foregoing  theorems  will  enable  us  to  find  the  integral 
of  any  entire  function  of  a  variable.  To  take  the  function  in 
its  most  general  form,  let  it  be  required  to  find  the  integral 

u—j  (ax"*  -[-  bx"^  -\-  ex'  -\-  »  .  ,)dx. 

By  Theorem  III., 

u=  I  aTf^dx  -{-  I  bx^dx  ■\-    I  cx'dx  -f-  o  o  •  . 


I  differeu' 


By  that 


lows. 


[x)dx. 


y  the  nega- 
Ue  derived 


I  of  several 
frals  of  the 


fzdx-\-  . . 

22. 

he  integral 
function  in 
3  integral 


TEE  ELEMENTARY  FORMS  OF  INTEGRATION.    205 


By  Theorem  II., 


/  ax^dx  =  a  I  x^dx; 
eic*  ere.  y 


and  by  Theorem  I., 


etc. 


w  +  1 
etc. 


By  successive  substitution  we  then  have 


u  = 


aa;"*+*    .  ia;"+*   .   cx^-*-^ 


--r  + 


+ 


-{■  ah^  -V  Ih^  4-  cA,  +  .  .  .  , 


m  -\-l      n  -{-1      jo  +  1 

where  /*,,  7i„  7i,,  etc.,  are  the  arbitrary  constants  added  to  tho 
separate  integrals. 

Since  the  sum  of  the  products  of  any  number  of  constants 
by  constant  factors  is  itself  a  constant,  we  may  represent  the 
sum  ah^  +  M,  -f  ^'*a  l>y  the  single  symbol  h.    Thus  we  have 


y  (aa;"*  -f  hx""  -f  cx^  -\-  .  .  .)dx 


ax 


m+l 


T  + 


Ja;"+' 


ex 


p+i 


w  +  1    '  n  +  1   '  p  \-l 

EXERCISES. 


-{-  ,  .  .-\-h. 


Form  the  integrals  of  the  following  expressions,  multiplied 
by  dx\ 


I.   X \ 


-t 


.-s 


S.  «a;'.  6.  Ja;'.  7.  ax~*,  8.  Ja;~'. 

9.  rtiB  +  I.       10.  «a;'  —  c.      II.  ax^  +  c.r.     12.  ax^  —  ca;* 
13.  a;*.  14.  xi.  15.  a;-*.  16.  ax-^. 

17.  «.T*— o.-c-*.  18.  7»a;* :.    10.  — .,      20,  a  A — ■«. 

a*  XX.  X 

120.  ^Ae  Logarithmic  Function,  An  exceptional  case 
of  Theorem  I.  occurs  when  n  —  —  1,  because  then  n  -{-1 
=  0,  and  the  function  becomes  infinite  in  form.     But  since 

d'\os  X  =  —  =  x~^dx, 
°  X 


^I'r 


f. 


i 

I 

j  I 

I, 


■  j  I 


!  I  M 


206 


THE  INTEaBAL  CALCULUS. 


■1.1 .1 


it  follows  that  we  have  for  this  special  case 

/  x-^dx  =   /  —  =  log  a;  +  ^*- 


(«) 


Let  c  be  the  number  of  which  h  is  the  logarithm.    We  then 
have 

log  x-\-h-=  log  X  +  log  c  =  log  ex. 

We  may  equally  suppose 

7i  =  —  log  c  =  log  -. 


Then 


X 


log  X  -{-h^  log  — . 


or 


Hence  we  may  write  either 

rdx      , 
y-=log.:., 

/dx      ,      X 
—  =  log  -; 
X  °  c  ' 


(*) 


c  being  an  arbitrary  constant. 

We  thus  have  the  principle:  The  arbitrary  constant  added 
to  a  logarithm  may  he  introduced  hy  multiplying  or  dividing 
hy  an  arbitrary  constant  the  number  whose  logarithm  is  ex- 
pressed. 

121.  We  may  derive  the  integral  (a)  directly  from  Theo- 
rem I.,  thus:  In  the  general  form 


//pn  +  l 
X'^dx  = —r  +  h 
w  4- 1 


+ 

let  us  determine  the  constant  h  by  the  condition  that  the  in- 
tegral shall  vanish  when  x  has  some  determinate  value  a. 
This  gives 


a 


n  +  l 


n  +1 


+  A  =  0; 


h=~ 


a 


r  +  l 


W  +  1 


Thus  the  integral  will  become 


/  x^'dx  = 


X 


n+l 


a 


n  +  l 


71  +  1 


m^^' 


THE  ELEMENTARY  FORMS  OF  INTEGRATION.     207 

in  which  a  takes  the  place  of  the  arbitrary  constant.  This 
expression  becomes  indeterminate  for  ?i  =  —  1.  But  in  this 
case  its  limit  is  found  by  §  71,  Ex.  5,  to  be  log  x  —  log  a. 
Thus  we  have 


/ 


x^^dz  =  log  X  —  log  a  =  log  -, 


as  before,  log  a  being  now  the  arbitrary  constant. 

122.  Exponential  Functions.     Since  we  have 

e?(rt*)  =  log  a .  a^dxy 
it  follows  that  we  have 


f  log  a. a'dx  =  a*  -f  h. 


or,  applying  Th.  II.,  §  119,  to  the  first  member  and  then  di- 
viding by  log  a, 

«*  +  /* 


/ 


a'dx  = 


log  a' 


which  we  may  write  in  the  form 


/ 


a'dx  = 


a' 


log  a 


+  h. 


h 


because  z is  itself  a  constant  which  we  may  represent  by  h, 

123.  The  Elementary  Forms  of  Integration.  There  is 
no  general  method  for  finding  the  integral  of  a  given  differen- 
tial. What  we  have  to  do,  when  possible,  is  to  reduce  the 
differential  to  some  form  in  which  we  can  recognize  it  as  the 
differential  of  a  known  function.  For  this  purpose  the  fol- 
lowing elementary  forms,  derived  by  differentiation,  should 
be  well  memorized  by  the  student.  We  first  write  the  prin- 
cipal known  differentials,  and  to  the  left  give  the  integral, 
found  by  reversing  the  process.  For  perspicuity  we  repeat 
the  forms  already  found,  and  we  omit  the  constants  of  in- 
tegration. 


,.,, 

«r 

■ 

iil 

il! 

1 

■HI 


III 


208 


THE  INTEGRAL  CALCULUS. 


M      :i 


?-f  I      ' 


■1   ! 


i  ! 


y 


»    •  • 


/ 


y'^dy       = 


_  y 


n  +  l 


W  +  1 


(1) 


% 


Jit 

J     y 


=  log  y.       (2) 


d'sin  ij       =  cos  ydy,  .  •  .ycos  ydy  = 


sm 


y- 


(3) 


'.'  d-coay       =  —  sin  ydy, 
• .  •  6?-  tan  y       =  sec'  ydy, 
'.'d-coty       ^  -  ^y 


/•■ 


t/ 


■/ 


sin  ydy  =  —  cos  y  (4) 


•        2  * 

Sin   y 


dy 

cos'  w 

dy 


—  tan  ?/.      (5) 


^''f^.     =-coty.(6) 


sm   y 


.  .  7  tan  w  , 

'.'d'secy       = "d?/, 

cos  y  *^ 

•.wZ-sin<-'>v  =  — ^--, 

dy 


/tan  ?/</?/ 
-^~-=«ecy.      (7) 


Vl- 


y 


'.'  d'QOB^~^^y  —  — 


Vl 


r 


d-tan^~^^v  = 


y 


__      dy 


•■•/7f^="°^'""^-(9) 


1  +  ?/ 


a> 


^•a» 


^ 


=  n^  log  rt<^?y, 
dy 


■f 


^/;/ 


1  +  J/ 


-,   =tan(-')f/.(10) 


ly        =: 


ft" 


log 


a 


(11) 


d  •  sin  h^~  ^h/  =  — — " — 


Vf+1 


dy 


^?*cosh^~%= 


=  sin  h<-»)y  =  log  (y  +  Vf-{- 1).  (12) 


«/ 


■/ 


i^f-  1 

dy 


Vy" 


=  =  cos  h^-')?/  =  log  (y  4-  i/y»-  1).  (13) 


c?*tanh^~*\y 


__      <fy 


2/ 


a> 


tanh^-"y  =  --  log 


2 


If  y 
1-y- 


(14) 


INTEGRALS  REDUCIBLE  TO  ELEMENTARY  FORMS.  209 


I    f 


CHAPTER  II. 

INTEGRALS    IMMEDIATELY.  REDUCIBLE    TO    THE 
ELEMENTARY  FORMS. 

124.  Integrals  Reducible  to  the  Form  I  y^dy.  The  fol- 
lowing are  examples  of  how,  by  suitable  transformations,  we 
may  reduce  integrals  to  the  form  (1).    Let  it  be  required  to  find 

i{a  4-  xYdx. 

We  might  develop  {a  +  xY  by  the  binomial  thorem,  and 
then  integrate  each  term  separately  by  applying  Theorem  III., 
§  119.  But  the  following  is  a  simpler  way.  Since  we  have 
dx  =  d{a  +  ^)>  we  may  write  the  integral  thus: 

/  {a  -f  xYd{a  +  x). 

It  is  now  in  the  form  (1),  y  being  replaced  hy  a  -\-  x. 
Hence 


(1) 


In  the  same  way, 
/  {a  —  xYdx  =  —  J  {a—  xYd{a  —  x)  =h  —       ~Z\ — • 
To  take  another  step,  let  us  have  to  find 

f{a  +  IxYdx. 


We  have 


1  1 

dx  =  ■j-d{hx)  —  jd{a  -(-  Ix), 


Hence,  by  applying  Th.  II., 


.1 


:.:■ 


7 


I  r 


11'  i; 


210 


THE  INTEGRAL  CALCULUS. 


i  « 


1  il 


We  might  also  introduce  a  nev/  symbol,  y  =a  -{-hx,  and 
then  we  should  have  to  integrate  y"^/y  with  the  result  in  §  123. 
Substituting  for  y  its  value  in  terms  of  x,  we  should  then  have 
the  result  (2).* 

These  transformations  apply  equally  whether  n,  a  and  b 
are  entire  or  f ractional,  positive  or  negative. 

EXERCISES. 

Find:  i.  /  («  -f-  xydx,        2.  /  d{a  —  xydx» 

3.  /  (rt  —  2xydx,      4.  I  {a-{-  x)''*dx,     $.  I  {a  —  x)~*dx, 

6.  /  (a-{-mx)~^dx.    y.  1  {a  —  mxydx.     8.  /  {a  —  mx)~^dx. 

/*     dx  f*     dx  p      dx 

9-y  («  ^  ^y  'o  y  (^  _  ^)»-  "-J  («  _  4a;)"^-    ' 

12.  /  (rt  -}-  xydx.      13.  /  (a  -f  nxydx.    14.  /  («  +  x^Yxdx. 

'^-/{^^  +  7'  +  ¥*)^^^-  ^^•/(^^-• 

19.  /  (a-{-bx  -\-  cx^){b  -\-  2cx)dx. 

20.  /  («-[-&«  +  ca;'')"(J  -f  2cx)dx, 


21 


■/ 


(a  +  /S'a;  +  crc*)* 


*  The  question  whether  to  introduce  a  new  symbol  for  a  function 
whose  difTerential  is  to  he  used  must  lie  decided  by  the  student  in  eacli 
case.  He  is  advised,  as  a  rule,  to  first  use  the  function,  because  he  then 
gets  a  clearer  view  of  the  nature  of  the  transformation.  He  can  then 
replace  the  function  by  a  new  symbol  whenever  the  labor  of  repeatedly 
writing  the  function  will  thereby  be  saved. 


IMi. 


INTEGRALS  REDUCIBLE  TO  ELEMENTARY  FORMS.  211 


125.  A2)plication  to  the  Case  of  a  Falling  Body.      We 

have  shown  (§  33)  that  if,  at  a  time  tj  a  body  is  at  a  distance 

z  from  a  point,  the  velocity  of  motion  of  the  body  is  equal  to 

dz 
the  derivative  --,     Now,  when  a  body  falls  from  a  height 

under  the  influence  of  a  uniform  force  g  of  gravity,  unmodi- 
fied by  any  resistance,  the  law  in  question  asserts  that  equal 
velocities  are  added  in  equal  times.  That  is,  if  z  be  tlio 
height  of  the  body  above  the  surface  of  the  earth,  and  if  we 
count  the  time  t  from  the  moment  at  which  the  body  began 
to  fall,  the  law  asserts  that 


dz  , 

It  =  -  ^*' 


(«) 


the  negative  sign  indicating  that  the  force  g  acts  so  as  to 
diminish  the  height  z. 
By  integrating  this  expression,  we  have 

z=h-  igt\  (b ) 

Here  the  constant  7i  represents  the  height  z  of  the  body  at 
the  moment  when  /  =  0,  or  when  the  body  began  to  fall. 

From  the  definition  of  h  and  z,  it  follows  that  h  —  z  ia  the 

distance  through  which  the  body  has  fallen.     The  equation 

{b)  gives 

h  —  z  =  ^gt^.  (c) 

Hence:  The  distance  through  which  the  body  has  fallen  is 
proportio7ial  to  the  square  of  the  time. 

At  the  end  of  the  time  t  the  velocity  of  the  body,  meas- 
ured downwards,  is,  by  {a),  equal  to  gt.  If  at  this  moment 
the  velocity  became  constant,  the  body  would,  in  another 
equal  interval  t,  move  through  the  space  gt  X  t  —  gt"^. 
Hence,  by  comparing  with  (c)  we  reach  by  another  method  a 
result  of  §  33,  namely: 

In  any  period  of  time  a  body  falls  from  a  state  of  rest 
through  half  the  distance  througn  which  it  would  move  in  the 
same  period  luith  its  acquired  velocity  at  the  end  of  the  period. 


1 
If 

I 


'Si 


ii 

fv 
<  ■ 

;      i       I 


;    1: 


r 


Ji 


■4\ 
m 


m  1 


;? 


ffll  't 


IH 


212 


riTi?  INTEGltAL  CALCULUS. 


136.  Reduction  to  the  Logarithmic  Form,    Let  ua  havo 
to  find 

/mdx 
ax  +  d* 


Since 


<fa;  =  -d{<ix)  =  -r/(aa;  +  b). 


wo  may  write  this  expression  in  the  form 

/m  d{ax  -f  i) 

and  the  integral  becomes 

—  ^  pd{ax  -\-  h)  _  w  .     ax-\-h 
~^aj     ax  -^-b    ~ "      ^  " 

c  being  an  arbitrary  constant. 


a 


EXERCISES. 

Integrate  tho  following  expressions  multiplied  by  dx\ 


I.  x-\--, 

X 


4»  — ; — 7* 
x-\-\ 

7. \-x\ 

nx 


I 

2.    — . 
X 


•'•  %x  -  r 

a 


m 
x 


6. 


m 


10. 


«"  +  ^rc 


4  +  ^* 
13.  Find/^ 


8. 


XI. 


ca;  —  b' 


a' 


2aa;  +  b' 
a-\-b 


7W  —  n 


xdx 


-»• 


/xdx 
r 


12. 


mx  —  n 


15. 


Note  that  iB  c/a;  =  ld{x')  =  |^/(1  +  a;'). 
xdx  ,       /» a;'  ^a; 


/a;aa;  /» i; 

a'  —  wa;'*  '  J  1 


+  ^' 


!?■ 


/ 


log  X  dx 


X 


dx 


Note  that  log  x  --  =  log  xd .  log  x. 


^log(l  +  y) 


xdx 


J       1  +  y      ^      ^    y    4/a»  4.  a;»  c'  (1  -  a;' 


(1  -  a:')^ 


INTEORAia  REDUCIBLE  TO  ELEMENTARY  FORMS.  213 


l!<i7.  Trigonometric  Forms,    Tho  following  aro  examples 
of  the  reduction  of  certain  trigonometric  forms: 

/COB  mz  dx=i  —  I  cos  mx  climx)  =  —  sin  mx  4-  h, 

/sin  mx  dx=^  —  (  sin  mx  dimx)  =  h cos  mx, 
mj                ^     '              m 

I  cos  (a  +  'nix)dx  =  -  /  cos  (rt  -f-  mx)d(a  +  inx) 

__  sin  (a  +  ;/«:?;)       , 
m 

/,          ,          psmxdx  pd'coax 

tan  xdx=    I =  —   / 
t/     cos  a;  J 


where  7t  =  log  c. 
In  the  same  way. 


cos  X  J     cos  a; 

=  h  —  log  cos  a;  =  log  c  sec  a;, 


«/  si 


d» 


sin  a;  cos  x 


I  cot  a;  rfa;  =  log  c  sin  a;. 

/I       <fa;    __    od'i&n  x 
tan  a;  cos'a;  ~"  J 


/dx    __  1   /» 
sin  a;  ~~  2 1/  sin  ^; 


tan  a; 


=  log  c  tan  a;. 


-  =  W  c  tan -a:. 
X  cos  ^a;         °  2 


/rfa;    _    /»  fl?a;  __ .  /nr      a:  \ 

cos  a; "~  ,/  sin  (^;r  —  a;)  ~     ^  \4       2/ 


EXERCISES. 


Integrate : 

I.  (1  +  cos  y)dy.  2.  (1  —  e  sin  u)du. 

3.  cos  2y  c?y.  ^ws.  ^  /  cos  2yd(2y)  =  ^  sin  2y. 

4.  8in2yd?y.  5.  cob  7iy  dy. 

6.  sin  y  cos  y  dy,  Ans.  \  I  sin  2yd{2y)  =  —  i  cos  2y. 

7.  tan  2a;  dx,  8,  cot  2a;  efa;. 
9.  2  cos'  a;  (fa;. 

Ans.    /  (1  +  cos  2a;)<7a;  =  a:  +  A  -|-  ^  sin  2a;. 


\   ( 


ih 


,■3 


•< 


.1 


u 


'"  1 


214 


TllK  INTEQUAL  CALCULUS. 


IN\ 


\\\'. 


\ 


10.  2  sin'  X  dx. 


II.  tan  2y  Jy. 


12. 


»3 


cos  y  tly       ,         P(ii\ 
1  -f  sm  V  t/      1 


/(I  -f-  sin  y)  _ 


-f  sm  y 
sin  y  ^/// 


+  sm7/ 


log  6(1  4-  sin  y). 


1  -|-  cos  y' 

cos ;/  dy 

1  —  sin  y 


sin  11  dy 

14  •'    ^ 


i6. 


cos'  .r  —  sin'  x 


sin  1^:^; 


dx. 


1 8. 


r/a; 


20. 


(fa; 


1  —  cos  y 

sin  2y  </y 

cos  y 

sin  2x 

fir 

cos'  a;  —  sin' 

X 

21.    -: — 

dx 

17. 

19. 

COS  mx  sm  mx  sin  ?/ia;  cos  mx 

22.  sin  (ma;  -f-  a)dx,  23.  cos  («  —  nx)dx. 

24,  tan  nxdx,  25.  tan  (2;«  —  «)(/a;. 

dx  dx  ^  dx 

27 


26. 

29. 


sin  (a  —  a;)' 

cos  nydy 

a  -\-  sin  ?iy' 


30 


cos  {b  —  9ia;)* 
sin  nydy 
a  —  cos  ny' 


28.  -^    .  .. 

sm  {a  —  i(x) 

sec'  a;<Za; 

a—  m  tan  a; 


138.  Integration  of  ^  ^  ^  and  -^ 


^a* 


a  -{-  X  a'  —  X' 

We  see  at  once  that  tin  first  differential  may  bo  reduced  to 
that  of  an  inverse  tangenc ;  thus, 


dx 


1      dx 


Hence 


X*  +  «'      ^*'  ^'  _i_  1 


d- 


1  4) 


"  ^  4-  1 


J  a  -\-x        a*y   X*  a  a 

a 

We  find  in  the  same  way 

f*     dx  1-1,  f-i)-^    ,7        1  1        a-{-x 

I  -, ;  =  -  tan  h  ^-"  -  -f  7i  =  --  log  c — ■ — , 

J  a  —  X       a  a  ^a  a  —  x 

0  being  an  arbitrary  constant  factor. 


(1) 


('0 


Tl 

chai 

I.l 

of  a| 

I] 


INTEGUALIS  UEDUCIBLE  TO  KLEMEJSTAUY  FOUMIS.   215 


X  cos  VIX 


— HI — TTTi' 

(I     y~  ox  ~J~  ex 

Tho  reduction  of  integrals  of  this  form  depends  upon  the 
character  of  the  roots  of  the  quadratic  equation 

ex*  ■\- bx -\- a  =z  0.  (1) 

I.  If  these  roots  arc  imaginary,  the  integral  is  tlie  inverse 
of  a  trigonometric  tangent. 

II.  If  the  roots  are  real  and  unequal,  tho  integral  is  tho 
inverse  of  an  hyperbolic  tangent. 

III.  If  the  roots  are  real  and  equal,  that  is,  if  the  above  ex- 
pression is  a  perfect  square,  the  integral  is  an  algebraic  frac- 
tion. 

Dividing  the  denominator  of  the  fraction  by  the  coefQcient 
of  a;',  the  given  integral  may  be  written 

dx 


^t/    , 


^  .    bx       a 
«  H h  - 

CO 


(a) 


Writing  3/j  for  -  and  q  for  -;-,  the  expression  to  be  into- 

C  6 


grated  may  be  reduced  to  one  of  the  forms  of  §  128,  thus: 
dx  _  ^^  _         ^^(^  +  P) 


(^) 


x^  +  %px  +  q    {x  +py  +  s'  -y     {x-^pY  +  q  -y 

The  three  cases  now  depend  on  the  sign  oi  q  —  p*. 

I.  If  q  —p^  is  positive,  the  roots  of  (1)  are  imaginary  and 
the  form  is  the  first  of  the  last  article  with  x-^p  in  the  place 
of  x,  and  q  —  p^  in  the  place  of  a'.     Ilence  we  have 

dx  n        d{x  -j-  p) 


f 


=  /-(. 


x^-\-2px-^q       J  (x-\-  pY  +  q-p' 

1         .      ,^,.     X  -\-  p 


tan(-« 


=  +  /^.     a) 


Comparing  this  result  with  (a),  we  see  that  this  integral 
may  be  reduced  to  its  primitive  form  by  changing  p  into 


1  1 '    I ; 


I ; 


lil 


216 


THE  INTEGRAL  CALCULUS, 


IN 


11 


1  h  a 

-  —  and  q  into  -.     Substituting  and  reducing,  we  have 

Z  c  c 


J  a 


dx 


+  ia;  +  cz 


'~^e/^. 


dx 


X  -{--X  A — 


1 

c 


x-^r 


tan^-" 


2c 


'^  c       4c'  '^  c       4c» 


2 


, tan<-> -I^^L  + /..        (2) 

II.  If  q  —p*  is  negative,  that  is,  if  4ac  — -  J"  is  negative  in 
(3),  the  expression  (2)  will  contain  two  imaginary  quantities. 
But  these  two  quantities  cancel  each  other,  so  that  the  ex- 
pression is  always  real.     "When  q  —  p*  is  negative,  we  write 

{h)  in  the  form 

ci{x-\-p) 

p'-q-{x-^py' 

The  integral  is  now  in  the  form  (2)  of  §  128,  and  we  have 

dx  n         d{x  -f-  p) 


s 


x^  +  2j3a:  +  q 


/'         a\x  - 
V^q- 


(^  +  VT 


=  A- 


=  h- 


Vp' 


=rtanh<-« 


x-^p 


Vjf  —  q 


log  ^Vf-q  +  x  +  p^  ^3^ 


^  Vy  —  q      °     ^P'—  1—{x-\-p) 


Making  the  same  substitutions  in  these  equations  that  we 
made  in  Case  I.,  we  find 

dx  ,  3         ,      ,  (_i)    2ca;  -f  ^ 


J  a 


-\-bx-\-  ex' 


=  7i  - 


=  h- 


1 


tan  h^" 


\^b''  —  ^ac 


i/l,^^iac-\-2cx-\-b  ,,. 

, -  log  c—-  -^—•(4) 

4/6'-4r?c  ^/b^-^ac-{2cxAry) 

III.  If  p^  —  q=.  0,  the  expression  to  be  integrated  becomes 

dx 
T— j— -T5.     We  have  already  integrated  this  form  and  found 

dx  ^  1 

n  — 


f 


i^+Pf 


x-^p 


I. 


INTEQliALS  REDUCIBLE  TO  ELEMENTARY  FORMS.  217 


r 


e  have 


h  h.        (2) 

Degative  in 

quantities. 

lat  the  ex- 

e,  we  write 


i  we  have 


-^+P.  (3) 
IS  that  we 


4iaG 

cx-{-b) 
i  becomes 

d  found 


EXERCISES. 


Integrate  the  following  expressions: 
dx  dx 


I. 


x^  -  2x  -  4* 
dx 


2. 


{x  -a){x  -fi)'     ^'  a-\-  2bx  -  a;'* 
dx  dx 


dx 


130.  Inverse  Sines  and  Cosines  as  Integrals.  From  what 
has  already  been  shown  (§  123,  (8)  and  (9)),  it  will  be  seen  that 
we  have  the  two  following  integral  forms: 


—  =  sin  ^~^>  a;  +  A  E  ui 
Vl-x' 

==  =  cos  ^~^^x-\-  li'  =  u'\ 


where  we  have  added  h  and  W  as  arbitrary  constants  of  in- 
tegration. 

Comparing  the  first  members  of  these  equations,  we  see 
that  each  is  the  negative  of  the  other.  The  question  may 
therefore  be  asked  why  we  should  not  write  the  second 
equation  in  the  form 

^•*  -lit     _•__  t—\\ ..  /  \ 


u 


=  -/- 


=  li"—  sin^'^^o;. 


yi-a;" 

as  well  as  in  the  form  (i).  The  answer  is  that  no  error 
would  arise  in  doing  so,  because  the  forms  (6)  and  (c)  are 
equivalent.     From  {b)  we  derive 

X  =  cos  {u'  —  V)  —  cos  (/*'  —  u')\  (d) 

and  from  (c), 

X  =  sin  {h"  -  u').  (e) 

Now,  we  always  have  sin  (a  +  90°)  =  cos  a.     Hence  (d) 
and  (e)  become  identical  by  putting 

A"  =  h'  +  90°, 

which  we  may  always  do,  because  the  value  of  A"  is  quite 
arbitrary. 


il  :: 


> :  I  i 

m 


1 1  r 


218 


THE  IJSTEGBAL  CALCULUS. 


131.  The  preceding  reasoning  illustrates  the  fact  that 
integrals  expressed  by  circular  functions  may  be  expressed 
either  in  the  direct  or  inverse  form.  That  is,  if  the  relation 
between  the  differentials  of  u  and  x  ifi  expressed  in  the  form 


du 


dx 


Vl 


X 


i> 


we  may  express  the  relation  between  u  and  x  themselves 
either  in  the  form 

u  =  Bm^~^'^ X  •\- h 
or  in  the  form  x  =  sin  (u  —  h). 

So,  also,  in  the  form  (1)  of  §  128  we  may  express  the  rela- 
tion between  x  and  i(>  either  as  it  is  there  written  or  in  the 
reverse  form, 

a;  =  a  tan  a(u  —  h). 


132.  Integration  of 


dx 


We  have 


dx 


i/a"  T  x' 


d'^- 


X' 


sin(-»)-  +  A. 
a 


(1) 


In  the  same  way 

f-7^^     =cos^-'>-4-^    or    A-sin(-^>-,     (2) 
^    Va'  -  a;'  «  a      ^  ' 

We  also  have 


=  log-(^  +  ^^'  +  «').(3) 


a 


INTEGRALS  REDUCIBLE  TO  ELEMENTARY  FORMS.  219 


I  fact  that 

I  expressed 

be  relation 

the  form 


themselves 


3  the  rela- 
L  or  in  the 


h. 


(1) 


^)-.      (2) 


+  «').(3) 


r--t=.  =   C-^^^  =  cos  h  <->  i  +  A 


=  \og-(x+V'.^-a').    (4) 


EXERCISES. 


Integrate  the  differentials: 
dx 


I. 


Vc  —  x' 

ndy 
Va'  -  nY 

mdz 


II. 


13. 


dy 

'   Via'  4-  V 
dy 


Vb-{-~c(x-ay 
2xdx 


Va*- 


2. 

4. 

6. 

8. 
10, 
12. 
14. 


rf«/ 


l/4« 

dx 

Va' 

-{X- 

dz 

ay 

Via 

'  -  m'z' 

mdx 

Va' 

2      2 

--  fU  X 

dx 

Va* 

4-  wi'(a; 
dx 

-ay 

V(x 

-  ay  - 

■ic' 

nx'' 

^-'dx 

a;' 


^^2u  _  ^2» 


COS  xdx 


15.  If  du  =  — -===r-nrr  then  ein  X  =  a  cos  (?<  +  h). 
Va  —  sm' «  \     I     / 


16. 


18. 


20. 


e*^:c 


Vl  -e 


,2x 


17. 


<?a; 


—  sin  xdx 
a*  -\-  cos'  x' 

{x  —  «T)rfa; 
1^1  -  (a;  -  ap 


19 


^V'l-Cloga;)' 
cos  xdx 


21. 


a'  +  sin'  x' 
(x  4-  «)^^ 

1/1  +  {x  +"«y 


^ii:; 


I  > 


iill, 


220 


THE  INTEGIUL  CALCULUS. 


183.  Integration  of 


clx 


Every  differential  of 


Va  -\-  bx  ±  ex'' 

this  form  can  be  reduced  to  one  of  the  three  forms  of  the 
preceding  article  by  a  process  similar  to  that  of  §  129.     The 
mode  of  reduction  will  depend  upon  the  sign  of  the  term  ax'. 
Case  I.   The  term  ex*  is  negative.    Putting,  as  before. 


_1  h 


a 


we  have 

V«  +  hx  -  ex'  =  Ve  Vq  +  '^j^x  -  a;'  =  V^  ^F^~^H-J^-p)' 
Then,  comparing  with  (1)  of  §  133,  wc  find 
dx  1     /*  d{x  —  p) 


r dx -  Jl  /*. 


Vc*^    Vp""  -\-  q  —  {x  —  pY 


=  — ==sm 


(-1) 


X 


P    _ 


V^ 


-zzBin 


(-1). 


2ex 


(1) 


Vp'  +  q        Vc  Vb'  +  lae 

In  order  that  this  expression  may  be  real,  p'^  -\-  q  or  b'  -\-  iae 
must  be  positive.  If  this  quantity  is  negative  the  integral 
will  be  wholly  imaginary,  but  may  be  reduced  to  an  inverse 
hyperbolic  sine  multiplied  by  the  imaginary  unit. 

Casi:  II.   The  term  ex'  is  p)Ositive.     We  now  have 

Va  -f  bx  -{-  ex'  —  Ve  V{x  -f-  p)'  -{-  q  ~  p"' 

dx  _    1     /*  ^^(-^  +  V\ 

^bx  +  ex'  ~   VcJ    ^{x+pY  -{-q  -  p' 


J  Va 


=  —  log  C{x  -}-p  +  Vx'  +  2px  +  q) 

re 


C 


=  T  log  ;7i(3ca;  +  b-i-  2c*  Va  +  bx  +  ex'). 

Because  Cis  an  arbitrary  quantity,  the  quotient  of  C  by 
2ei  is  equally  an  arbitrary  quantity,  and  may  be  represented 
by  the  single  symbol  C.     Thus  we  have 


/ 


dx 

Va~j^bx  + 


^  =  —log  C{b-{-2ex-^2  Vc  Va-{-bx-\-cx').{2) 


ex 


INTEGRALS  REDUCIBLE  TO  ELEMENTARY  FORMS.  221 


M! 


(rential  of 

IS  of  the 
29.  The 
term  cx^. 
(fore, 


-  ix-p)\ 


b_ 

" —  • 


(1) 


rb'  +  4:ac 
Q  integral 
,n  inverse 


\-q) 


bx  -\-  cx^). 

of  C  by 
presented 


+.:.').  (2) 


Integrate: 


EXERCISES. 


dv 


I. 


V^a"  +  4:by  —  f 

ydy    _ 

cos  Odd 


7. 


sin  d  —  sin'  0 
sin  6  cos  Odd 


V4:  -  COS  26*  -  cos''  20 


2. 

4. 
6. 
8. 


(?y 


|/(^  +  ?/)  (^»  -  y) 

dy 

VaY  -  by  -{-  y' 
cos  OdO 

a  sin  ^rZ/9 


Va'  -  Z>"(i  -  CCS  0) 


■i' 


134.  Exponential  Formfi.  Using  the  form  (11)  of  §  123, 
we  may  reduce  and  integrate  the  simplest  exponential  dif- 
ferentials as  follows: 

a'^''dx  =  ~  /  « «* d(mx)  =  —, h  h.  (1) 

jnj  ^  m  log  a  ^  ' 

Ca'+^'dx  =    fa^-^^dix  -{- b)  =  ~-  +  h,  (2) 

/I      /»                                                      ^  ma;  +  6 
^mx  +  6^;^^  ±    /  a'»*  +  ^/(wX+3)  =-^i [-h,      (3) 
7Wt/                       ^                   '         ?>ilog«  ^    ' 

Ca-'^'dx  ■= Ca-'^^'di^—  mx)  =  -^;:77i;^— -•       (4) 


in  log  «  * 


Integrate: 
r.  e^dx. 
4.  {a  4-  b)e''dx. 


EXERCISES. 

2.  b^dy. 
5.  a^-^dy. 


3.  a^-'^dy. 
6.  a~'^dx. 


7.  («^  + rt-*)fZa;.       8.   (a*  —  rt-'')^Zaj.       9.  (rt  +  e'')r7a;. 


10.  (a^''—a~'^)dx.     II. 


e^^f?.?: 


1  -I-  e*' 
14.  (1  +  a'^ydx. 


12. 


e^'^dx 
1+7'^* 


15.  («"^  +  «-""«)'t/a;.  16.     Ce^^xdx. 


17 


•/ 


e^^'^xdx. 


[8.     /*e  -  «(^' -  %c?a;. 


I, 


<  ; 


|i! 


i     ! 


^r 


;   ! 


li 


\i  :il' 


in 


::!K 


vim 


■•it 


ii  H!! 


r 


11 


■M 


222 


2!ffiE?  INTEGRAL  CALCULUS. 


CHAPTER  III. 

INTEGRATION   BY  RATIONAL  TRANSFORMATIONS. 

135.  We  haye  now  to  consider  certain  forms  which  cannot 
be  reduced  so  simply  and  directly  as  those  treated  in  the  last 
chapter.  Before  passing  to  general  methods  we  shall  consider 
some  simple  cases. 

I.  Integration  of ^^^—dx.  Any  form  of  this  kind,  when 

m  is  entire,  may  be  integrated  by  developing  the  numerator 
by  the  binomial  theorem.     We  then  have  1 


{a  +  xY  _  ^  I 


ma  •"  ~  ^« 


+ 


e)«-^'+eto.. 


and  each  term  can  be  integrated  separately.     \in  <  in  •\-  2, 
and  entire,  one  of  the  terms  of  the  integral  will  contain  log  x. 


II.  Integration  of 


x^dx 


We  may  reduce  this  form  to 


{a  +  bx)*"' 

the  preceding,  by  introducing  a  new  variable,  z,  defined  by 

the  equation 

z  =  a-{-  bx, 

rrii  .      .  z  —  a      ,        dz 

This  gives 


X  = 


dx  = 


b     ' b' 

Substituting  these  values  of  x  and  dx  in  the  expression  to 
be  integrated,  it  becomes 

{z  —  a)^dz 

wh'.ch  may  be  integrated  by  the  method  of  the  last  article. 

III.  Integration  of  — -t-t — r — a* 

a  "T*  ox  "jc,  cx 


We  reduce  the  denomi- 


1ATI0NS. 

lich  cannot 
in  the  last 
Al  consider 


cind,  when 
numerator 


INTEGRATION  BY  RATIONAL  TRANSFORMATIONS.    223 

nator  to  the  form  ±  (y  -  q)  ±  {x  +  jw)'  as  in  §  129.     Then, 
putting,  for  brevity, 

i'  =y  -  q, 

z-=x-\-p, 
which  gives  dx  =  dz, 

the  integration  will  have  to  be  performed  on  an  expression  of 

the  form 

{z  —  p)dx  __    zdz  pdz 

b'  ±  z'"  "  ¥~±~?  ~  F±7'' 

Each  of  these  terms  may  be  integrated  by  methods  already 
given  (§§  126,  128). 
The  process  is  exactly  the  same  if  we  have  to  find 

{a  +  bx)dx 


N 


±{^-pr 


be, 

<  m  +  2, 

tain  log  X. 

lis  form  to 
defined  by 


EXERCISES. 


)ression  to 


article, 
e  denomi- 


Integrate: 
(x  —  aydx 

1 .  5  • 

X 

x*dx 

dx 

\a      XI 
x^dx 


II. 


•  (a'  -  xY 
xdx 

*  '^'^{b-xf 
(y  -  b)dy 


13 


{y-f^y  +  {y  +  hr 

{x  —  a)dx 


15 


:c{pi,  —  b)  ' 
z'^dz 


2. 


(1  -  1)V. 

\a       xj 


X 

x^dx 


^'  (1+^:7* 

,    (x  -\-  a)dx 
o. 

8. 


{a  -  xy  • 

x^dx 


lO. 


(1  _  ir 

W       x'  I 

zdz 


12. 

i6. 


{a  +  zY  +  (a  -  zy 

{z  —  c)dz 
a^  —  az  -\-  z^' 

{y  +  a)dy 
'a^-{y  +  br 

z^dz 
(1  -  zf 


•'  t 


lilt 


:i 


n 


I  i 


I .  ■ 


S  I 


f !  ; 


Ui 


224 


THE  INTEOHAL  CALCULUS. 


1  i 


il!   'm 


136.  Rcductio7i  of  Rational  Fractions  in  general,  A  ra- 
tional fraction  is  a  fraction  whose  numerator  and  denominator 
are  entire  functions  of  the  variable.     The  general  form  is 

q.  +  (7,^  -f  q^^"  +  .  .  .  +  qn^""  ~  D ' 

If  the  degree  w  of  the  numerator  exceeds  the  degree  n  of 
the  denominator,  we  may  divide  the  numerator  by  the  de- 
nominator until  we  have  a  remainder  of  less  degree  than  n. 
Then,  if  we  put  Q  for  the  entire  part  of  the  quotient,  and  R 
for  the  remainder,  the  fraction  will  be  reduced  to 


D       ^^  D' 
If  we  have  to  integrate  this  expression,  then,  since  Q  is  an 
entire  function  of  Xy  the  differential  Qdx  can  be  integrated 

by  §  119,  leaving  only  the  proper  fraction  -j-.     Now,  such  a 

fraction  always  admits  of  being  divided  into  the  sum  of  a 
series  of  partial  fractions  with  constant  numerators,  provided 
that  we  can  find  the  roots  of  the  equation  D  =  0.  The  theory 
of  this  process  belongs  to  Algebra,  but  we  shall  show  by  ex- 
amples how  to  execute  it  in  the  three  principal  cases  which 
may  arise. 

Case  I.  The  roots  of  the  equation  D  =  0  all  real  and  un- 
eqiial.  Let  these  roots  hQ  a,  ^,  y  .  ,  .  6.  Then,  as  shown 
in  Algebra,  we  shall  have 

D={x-  a){x  -  /3){x  -y)  .  .  .  (x-  6). 

"We  then  assume 

^  A      .      B      .       C 

D 


X 


a 


+ 


+  . 


•  •  > 


X  —  fi       X  —  y 

Ay  By  C,  etc.,  being  undetermined  coefficients.  To  deter- 
mine them  we  reduce  the  fractions  in  the  second  member  to 
the  common  denominator  D,  equate  the  sum  of  the  numera- 
tors of  the  new  fractions  to  Ry  and  then  equate  the  co- 
eflficients  of  like  powers  of  av 


iL  A  ra- 
lominator 
>rm  is 


jgrea  n  of 
y  the  de- 
)  than  71. 
it,  and  M 


e  ^  is  an 
ntegrated 

w,  such  a 

mm  of  a 
provided 
le  theory 
w  by  ex- 
es which 

and  un- 
is  shown 


0  deter- 

mber  to 

nuraera- 

the  co- 


INTEGRATION  BY  JIATIONAL  TRANSFORM iTIONS.    "22^) 
As  an  example,  lot  us  take  the  fraction 

■  ,    -  -  ax. 

X    —  X 

We  readily  find,  by  solving  the  equation  re'  —  a;  =  0, 


X 


.» 


Assume 

a; +  3 


X  =  x{x  -  l){x  +  1). 


X' 


=4+ 


B      .      C 


X  X  X  —  \         X  ■\-\ 

_  {A  J^B-\-C)x'  ^{D-  C)x-A 

—  IT  • 

X    —  X 

Equating  the  coefficients  of  poAvers  of  x,  we  have 

A-\-  B^  C-0) 
B-  C=l; 
A  =  -3; 
whence  B  =  2  and  0  =  1.     Hence 

a;  -f  •  3  _       3  2  1 

x"  —  x~       XX  —  Ix-^V 

and  then,  by  §  120, 

*J    X.    —  X  d      X  t/    X  —  1         d    X  -\-l 

=  -  3  log  a;  +  2  log  (x  -  1)  +  log  (a;  +  1)  +  log  C 
C{x  +  l){x  -  1)' 


log 


Integrate: 


X' 


EXERCISES. 


I. 

3- 

5- 
7 


{x  —  Vjdx 
X?  —  a;  —  6* 

xdx 
X^^x^' 
'a;'  4-  a;  +  1 
x^  -\-x^  —  %x 
{x"  +  2a;*)r?a; 


xdx 


2. 


x" 


r 


•  x"  +  2:c'  -  8a;* 
x^dx 
^*  a;'  -  {ci  ^h)x  +  ab' 


4 

6. 

8. 

lO. 


{x  +  x^yix 


'   {x-l){xi-l){x-2){x-\-2y 
x^dx 


9  a* 

X  —  a 


{x'  +  x')dx 


x{x  -  l){x  -1-  l)(a;  -  2)' 

dx 
x'  —  {a  +  hyx"  +  ahx' 


; 

ij  - 

! 

.  n 


i   ■:  \ 


1  '■ 


11 


m; 


'■■\^- 


ii 


h  i 


V'M 


226 


THE  INTEGRAL  CALCULUS. 


Case  II,  Soine  of  the  root  a  equal  to  mch  other.  Let  the 
factor  X  —  a  appear  in  D  to  the  nth.  power.  Then,  if  we 
followed  the  process  of  Case  I.,  we  should  find  ourselves  with 
more  equations  than  unknown  quantities,  because  the  n 
fractions 


X  —  a 


+ 


B 


X  —  a 


+ 


C 


X  —  a 


+  ... 


would  coalesce  into  one.     To  avoid  this  we  write  the  assumed 
series  of  fractions  in  the  form 


rn  + 


B 


n-1  "T"  •   •    •   + 


F 


X 


a 


4- 


H 


x—fi 


+  etc.. 


{x  —  a)"       {x  —  a) 

and  then  we  proceed  to  reduce  to  a  common  denominator  as 
before.     The  coefficients  Ay  B,  etc.,  are  now  equal  in  num- 
ber to  the  terms  of  the  equation  Z^  =  0,  so  that  we  shall  ha>e 
exactly  conditions  enough  to  determine  them. 
As  an  example,  let  it  be  required  to  integrate 

a;' -5  ^ 

X    —  X    —  X  -\-l 

We  have   x'  -  x'  -  x -{-1  -  (x  -  1)'  (x  +  1). 
We  then  assume 

x'  -  T)    _ A B  C 

{x-iy{x-{-i)~  {x-iy'^  x-i'^  x^i 

-  (g  +  Oy  +  {A  -  '^C)x  -{-A-B+C 

{x-iy{x-\-i) 

We  find,  by  equating  and  solving, 

A  =  -2; 
B  =  +2; 
C=-l, 
Hence 

x'-6  -  2 


i  + 


{x - iy{x -^1)" {x-iy^x-i    a:  +  r 


IN 
Tl 
-  2 


I. 


hi  \i 


Let  the 
n,  if  we 
Ives  with 
0  the  w 


INTEOBATION  BT  RATIONAL  TRANSFORMATIONS.    ^27 
The  required  integral  is 


-./(.-i)-V.  +  ./^-|L^-/,^^ 


2 


+  2  log  (.r  -  1)  -  log  (a:  +  1)  +  log  C 


x-l 

2       ,    .      C{x- 1)' 


^ 

1 

. 

1 

1 

1 

(11 


assumed 


-  etc., 

nator  as 
in  num- 
all  ha\e 


B+C 


EXERCISES. 


Integrate: 


I. 


dx 


x{x-]-iy' 

x^dx 


2. 


dx 


x\x  -  ly' 

dx 


^*  {x  -  iy{x  4-  ny 

{a  -f  x)dx 
^'  x\x  -  af 


"  {x  -  a)\x  -  by 


6. 


{a  —  x)dx 


x\x  -f-  ay{x  -  by 


Case  III.  Imaginary  roots.  Were  the  preceding  methods 
applied  without  change  to  the  case  when  the  equation  D  =  0 
has  imaginary  roots,  we  should  have  a  result  in  an  imaginary 
form,  though  actually  the  integral  is  real.  We  therefore 
modify  the  process  as  follows: 

It  is  shown  in  Algebra  that  imaginary  roots  enter  an  equa- 
tion in  pairs,  so  that  if  x  =  a  -{-  ^i  (where  i  E  V  —  1)  is  a 
root,  then  x  =  a  —  fti  will  be  another  root.  To  these  roots 
correspond  the  product 

(x-  a-  /3i)(x  -  «  +  /3i)  ={x-  ay  +  yS'. 

By  thus  combining  the  imaginary  factors  the  function  D 
will  be  divided  into  factors  all  of  which  are  real,  but  some 
of  which,  in  the  case  of  imaginary  roots,  will  bo  of  the  second 
degree. 

The  assumed  fraction  corresponding  to  a  pair  of  imaginary 
roots  we  place  in  the  form 

A-\-Bx 

{x  -  ay  +  /3" 


II 


I 


i! 

.1 1 


:■-:, 


'  s 


^28 


THE  INTKGRAL  CALCULm^. 


and  then  proceed  to  determine  A  and  B  as  before  by  equa- 
tions of  condition.  Wo  then  divide  the  numerator  A  +  Bx 
into  tlio  two  parts 

A  -}-  Ba    and     B{x  —  a), 

the  sum  of  which  is  vl  -f-  Bx.     Thus  we  have  to  integrate 


r     A-^Bcx  r  B(x  -  a)dx 

J  (a;  _  ay  +  /^'       "^  J  {x  -  ay  +  /i'* 

Tiio  first  term  of  (a)  is,  by  methods  already  developed, 

A  4-  Ba  .      ,   ..X  —  a 
____  tan<->^-, 

and  the  second  is 

^Blog{{x-ay-\-^^). 

"We  therefore  have,  for  the  complete  integral, 

r A^-\-Bx 

J  (x  - 


i") 


(.,  _  ay  +  p^ 


A  4-  Ba  ,      ,  ..X  —  a 
-jr —  tan  ^~  ^' 


+  i^  log!  (2: -«)'  +  /?•}+ A. 


i ' 


EXERCISES. 


IM 


'i     M 


rx-\-^x\ 

[.     /  — T — -T-dx. 
«/     :/;   —  1 


r  dx 


The  real  factors  of  the  denominator  in  Ex.  1  are  (a'  -\-  \){x  -f- 1)(«  —  1). 
We  resolve  the  given  fraction  in  the  form 

A-\-Bx  .      G      ,      D 


cc'  +  l     'aj  +  l'a;-l' 

X               1              \ 
and  find  it  equal  to  r  -| — -r—  -\ .    Then  the  integral  is  found 

to  be  \  log  (a;2  +  1)  +  log  (a;»  -  1). 

The  factors  of  the  denominator  in  Ex.  2  are  x—\  and  a;' -j- *  +  ^  ~ 
(«  +  i)'  +  f. 


+  1'  "*  t-/  x^  -  2x  +  4* 


r  dx 

''  J  x'+l'  ' 

Note  that  a;  +  2  is  a  factor  of  the  denominator  in  (4). 


JNTbJG RATION  BY  UATWNAL  THAN S FORMATIONS.    i?*iO 


e  by  equa- 
-iv  A-\-  Bx 


tegrate 


(") 


loped, 


n+h. 


f  1)(«  - 1). 


•al  is  found 
+  «+!  = 


137.   Integration  bif  Parts.  Lot  u  aud  v  be  any  two 
functions  of  x.     Wc  liavo 

lUiir)         (Iv  ,     (111 
ax           dx         ax 


By  transposing  iitid  iutegniting  wo  have 
(Iv  ,  /•  (In 


(1) 


which  is  a  general  formula  of  the  widest  iippllcation,  and 
should  bo  thoroughly  memorized  by  the  student.  It  shows 
us  that  whenever  the  dilTerential  function  to  be  integrated 

can  be  divided  into  two  factors,  one  of  which  (,//.')  can  bo 
integrated  by  itself,  the  problem  can  be  reduced  to  the  inte- 
gration of  some  new  expression  [v-j—dA. 

The  formula  may  bo  written  and  memorized  in  the  simpler 
form 

/  udv  —  iiv  —  /  vdUj  (2) 

it  being  understood  that  the  expressions  dv  and  du  mean  dif- 
ferentials with  respect  to  the  independent  variable,  whatever 
that  may  be. 

It  does  not  follow  that  the  new  expression  will  be  any  easier 
to  integrate  than  the  original  one;  aud  when  it  is  not,  the 
method  of  integrating  by  parts  will  not  lead  us  to  the  integral. 
The  cases  in  which  it  is  applicable  can  only  be  found  by  trial. 

The  general  rule  embodied  in  the  formula)  (1)  trnd  (•■i)  is 
this: 

Express  the  given  differential  as  the  product  of  o7iof  unction 
into  the  differential  of  a  second  function. 

Then  its  integral  toill  be  the  product  of  these  tivo  functions, 
minus  the  integral  of  the  second  fimction  into  the  differential 
of  the  first. 


I 


' 


;  ( 


« '\ 


lit 


m 


230 


TEE  INTEGRAL  CALCULUS. 


EXAMPLES   AND   EXERCISES  IN   INTEGRATION   BY   PARTS. 


JT 


I.  To  integrate  x  cos  xdx. 


Wo  have 


cos  xdx  =  d'&m  x. 


Therefore  in  (2)  we  have 

u^x\    v  =  sin  x; 
and  the  formula  becomes 

/  X  cos  xdx  =  I  xd'Bm  x  =  x  o'lux  —  I  mi  xdx 

=  2;  sin  a;  +  cos  a:  -|-  h, 
which  is  the  required  expressi^  a,  as  we  may  readily  prove  by 


differentiation. 
Show  in  ^he  same  way  that — 

r  ■  ^ 

2.  I  X  sin  xdx  =  —  x  cos  x  -\-  sin  x  -f-  h. 

3.  j  X  sec'  xdx  =  X  tan  x  —  (what  ?). 

4.  j  x  sin  X  cos  xdx  ■=.  —  \x  cos  "%%  +  i  sin  2.c  +  A. 

5.  /  log  xdx  -~  X  log  X  —  I  xd' log x~x  log  a;  —  a;  -f-  ^-^^ 

6.  The  process  in  question  may  be  applied  any  number  of 
times  in  succession.     For  example, 

I  X*  cos  xdx  =  /  re' c?"  sin  a;  =  x*  sin  x  —  %  j  x  sin  xdx. 

Then,  by  integrating  the  last  term  by  parts,  which  we  hiive 
already  done, 

/  x*  cos  xdx  =  a:'  sin  x  -\-'^x  cos  a;  —  2  sin  a;  +  ^• 

7.  In  the  same  way, 

/  a:'  cos  xdx  =        /  a:*c?*sin  a;  =      a:*  sin  a;  —  3  /  a:'  sin  x^/j; 
/  x'  sin  a;f?a;  =  —  /  a;'c?'  cos  a;  =  —  a;'  cos  x-\-%  I  x  cos  a;(?x. 


Y   PARTS. 


xdx 
proTe  by 


1u 

umber  of 

II  xdx. 
we  have 


sin  xdx; 
cog  xdx. 


INTEGRATION  BT  RATIONAL  TRANSFORMATIONS.    231 
Then,  by  substitution, 

/  x*  cos  xdx  =  {x*  —  Qx)  sin  J  +  (3a;'  —  6)  cos  x  +  h, 

8.  In  general, 

/  a;**  cos  xdx  =  /  x'^d'Bm x  =  x^BiD.x  —  n  / x"*'^  sin  xdx; 

—  I  x^~^  sin  xdx  =  /  x'*~^d'GOB  x 

—  a;"-*  cos  x  —  (n  —  1)  I  a;""^  cos  xdx; 

—  I  a;""'  cos  xdx  =—  aj^'^sin  a;  +(w  —  2)  /  a;**"'  sin  a;(?a;; 

/a;'*"'  sin  a;t?a;  =—  a^^-'cosa;  -{-(/j  —  3)  /a;""*  cos a;(/a;. 

etc.  etc.  etc. 

Then,  by  successive  substitution,  W'9  find,  for  the  required 
integral, 

\x''-n{n  -l)x''-^-^7i{n-l){n-%){n-^)x''-^— .  .  . }  sin  a; 
+  |wa;"-^  —  n(n  —  l){n  —  2)a;"-'+  .  .  .}  cosa;. 

9.  In  the  same  way,  show  that 

/  x^  sin  xdx  = 

I  -a:»+w(w-l)a;*-=*-w(n-l)(w-2)(w-3)a;'»-*+. . .  {  cosa: 
-}-  {^ia;*-^  —  7i(n  —  1)  (?i  —  3)a;'*-^  +•  •  •  |  sin  x. 

/I        p  x^  "*"* 

a;**  log  xdx  =  /  log  a:c?-  (3;"  +  ^)  =  -  log  a; 


1       /»x 
~  n  +  ij  ~ 


n+l 


-c?a;  = 


X 


n  +  l 


w  +  1 


log  a;  — 


X 


n+l 


II.,  Cxe-'^dx--  /*la:^-(e-««)  =  _^^f— I'+l  Ce-'^dx, 
J  J  a       ^        '  a         aj 

Now,  we  have    ie'^^dx  =  ■ 


,—  ax. 


Hence 


/ 


xe~'^dx  =  — 


xe 


—  ax 


a 


a 


s    • 


i'l 


'11 


;  ( 


232 


THE  INTEGBAL  CALCULUS. 


•   •% 


til  -u  /■', 


12.  To  integrate  x'^e~'^dx  when  m  is  a  positive  integer,  we 
proceed  in  the  same  way,  and  repeat  the  process  until  we  re- 
duce the  exponent  of  x  to  unity.     Thus, 


y.mn  —  <l^ 


x^e-'^dx  = \-  -  rx'^-h-'^dx, 

a  aj 

Treating  this  last  integral  in  the  same  way,  and  repeating 
the  process,  the  integral  becomes 


X^Q-ax         ^^m-lg-oa;         m{m  —  lyx'^'h' 


■  ax 


a 


a 


a 


—  etc. 


■ax 


a 


m+l 


{arx'^-\-ma'^-'^x'^-'^-\-m{ni—l)a'^^x'^*-\- . .  .+m!). 


13.  From  the  result  of  Ex.  5  show  that 

f{\og  xYdx  =  x{r  -2^  +  2)  -\-h, 

where  we  put,  for  brevity,  I  =  log  x. 

14.  Show  that,  in  general,  if  we  put 

u^  =y  (log  x)""  dx, 
then  iin  =  xl^  —  nu,^_i; 

and  therefore,  by  successive  substitution, 
w„  =  xO''  -  nl""-^  +  7i{n  -  1)V'-^  -  . 

15.  Deduce  (w  +  1)  C Px'^'dx  =  Fx"'  +  ^  -J'x'^  +  'dF. 

16.  Show  that  if  /  Pdx  =  Q, 

then  /  Px^'dx  =  Qx"*  —  n  I  Qx^-^dx. 

Also,  if  we  have 

/  Qdx   -  R\        I  Rdx  =  S,  etc., 


±  iil)  +  h. 


then 


fPx''dx  -  Qx""  -  nRx""-^  -\-  n{)i  -  1)/S'a;" 


etc. 


iteger,  we 
itil  we  re- 


r. 


repeating 


. .  .+m!). 


INTEGRATION  OF  IRRATIONAL  DIFFERENTIALS.  233 


CHAPTER  IV. 

INTEGRATION    OF   IRRATIONAL  ALGEBRAIC 
DIFFERENTIALS. 

138.  When  Fractional  Powers  of  the  Inde2yendent  Van- 
able  enter  into  the  Fxpressio7i.  In  this  case  we  may  render 
t'.ie  expression  rational  by  reducing  the  exponents  to  their 
least  common  denominator,  and  equating  the  variable  to  a  new 
variable  raised  to  the  power  represented  by  this  denominator. 

Example.     If  we  have  to  integrate 

—     '  .dx. 

1  +  a;* 

then,  the  L.  C.  D.  of  the  denominators  of  the  exponents  being 
6,  we  substitute  for  x  the  new  variable  z  determined  by  the 
equation 

X  =  z', 

which  gives  dx  =  Qz^dz, 

The  differential  expression  now  reduces  to 

^z'-{-z' 


z'  +  l 


dz. 


By  division  this  reduces  to 


6(/ 


,   ,     ,  ,x  7     ,     Gzdz     ,      Gdz 

z'  -\-z'  +  Z'  -  z  -  l)dz  +  ^^--  + 


/-f  1    '    ^"  +  1* 
Integrating  and  replacing  z  by  its  equivalent,  .<•*,  we  find 


/ 


— - — -,dx  =  ^x" 


-  etc. 


-f  3  log  (u:*-f  1)  4-  C  tan(-'>  x^  -j-  h 


1fl 
i:  \ 

ill 


:l: 


•  H 


iii 


i    '< 


¥ 


n 


m 


«! 


t-  i 


234 


THE  INTEGRAL  CALCULUS. 


If  the  fractional  exponent  belongs  to  a  function  of  x  of  the 
first  degree,  that  is,  of  the  form  ax  -\-  J,  we  apply  the  same 
method  by  substituting  the  new  variable  for  the  proper  root 
of  this  function. 

Example.    To  integrate 

{a  +  hx)^dx 
1  +  (a  +  bx)' 

We  put  (a  -f-  boc)^  =  z;    a  -{-bx  =  z^; 

2zdz 


dx  = 


b  ' 


{^'  -  ?^)' 


The  expression  to  be  integrated  now  becomes 

2z^dz     _2 
b{l  -{-z')~  b 

of  which  the  integral  is 

~(z-  tan<-»>2  +  A)  =  ^  I  {a  +  Jo;) *- tan <"»>(«  +  5^^)*+/*  I  • 


EXERCISES. 


Integrate: 
x^dx 


I. 


1-i-x 

(a  —  x)^dx 
l-\-a  —  x 

(x  +  c)*  , 
7.  ^^ — ■ — '—dx. 

(x^cf 

1  +  (^  -  c)\ 
l  +  (z-  c)^ 


2,    - 


*^a 


x^ax 


y 


5. 
8. 


1  +  a;^ 

{a  —  x)^dx 
I -(a-  xf 

{x  -  cf 
(x  — c)i 


3. xdx. 


6. 


dx. 


l±±^^dx. 
(a  -  x)^ 

(2x  —  a)^dx 


1  +  (2a;  -  a) 


1* 


10. 


dz. 


II. 


12. —ax, 

1- 


1  +  (a;  +  «)' 
=^*  +  IS 


^3. 


14 


^y  -{x-  «)* 


a:*- 


«?a;. 


»' 


{x-ay-\-{x-a)^ 


dx. 


INTEOBATION  OF  IRRATIONAL  DIFFERENTIALS.  235 


3f  X  of  the 

the  same 

roper  root 


rxY+h  \ . 


-xdx. 


t  —  X  , 
x)^ 

—  ayclx 
2x  —  a) 


i' 


139.  Cases  when  the   Given    Differential    contains   an 
Irrational  Quantity  of  the  Form 


Va-{-hx  -\-  ex*. 

It  is  a  fundamental  theorem  of  the  Integral  Calculus  that 
if  we  put  R  =  any  quadratic  function  of  x,  then  every  ex- 
pression of  the  form 

F{xy  VR)dx, 

(F{x,  VTl)  being  a  ratio7inl  function  of  x  and  ^R),  admits  of 
integration  in  terms  of  algebraic,  logarithmic,  trigonometric 
or  circular  functions.  But  if  R  contains  terms  of  the  third 
or  any  higher  order  in  x,  then  the  integ  :al  can,  in  general,  be 
expressed  only  in  terms  oi  .  ertain  higher  transcendent  func- 
tions know  as  elliptic  and  Abelian  functions. 

We  have  three  cases  of  a  quadratic  function  of  x. 

First  Case :  c  positive.  If  c  is  positive,  we  may  render  the 
expression  rational  by  substituting  for  x  the  variable  z,  de- 
termined by  the  equation 


Va-{-bx-\-  ex*  =  Vc(x  +  z)\ 

,  a -\- hx -{•  cx^  =  ex*  +  '^cxz  -f-  cz*. 


This  giv33 


X  = 


cz  —  a 


2cz' 


dx  =  —  2c 


a  —  Iz  -\-  cz* 
(b  -  2c'zy 


dz; 


Va  +  iz  +  cz'  =  -  iTe^L^hp^. 

0  —  2cz 


(«) 


By  substituting  the  values  givsn  by  (a),  (b)  and  (c)  for  the 
radical,  x,  and  dx,  the  expression  to  be  integrated  will  become 
rational. 

Second  Case :  a  positive  and  c  negative.  If  the  term  in  x* 
is  negative  while  a  is  positive,  we  put 

Va  +  bx 


We  thus  derive 


X  = 


,  ^.^.a  —    4/^  _|.  xz^ 

b-2Vaz 


(«) 


t 


11 


I  ; 


III 

ii.. 

I 

m 


f  i 


236 


TEE  INTEGRAL  CALCULUS. 


t  I 


'i 


¥!3 


dx  = 


_  2(  Vaz^  -Vac-  hz) 


Va-i-hx  —  cx^  =  — 


Vaz'  -  Va. 


ac 


Iz 


z'  +  c 


w 


io) 


The  substitution  of  Lliese  expressions  will  render  the  equa- 
tion to  he  integrated  rational. 

Third  Case :  a  and  c  both  negative.  If  the  extreme  terms 
of  the  trinomial  are  both  negative,  we  find  the  roots  of  the 
quadratic  equation 

—  a  -{■  bx  —  ex*  =  0, 
which  roots  we  call  a  and  /3.    We  then  have 

—  a-\-bx  —  ex*  =  c(a  —  x)  {x  —  /?), 
and  we  introduce  the  new  variable  z  by  the  condition 
V—  a-{-  bx  —  ex'  =  Vc(a  —  x)  {x  —  fi)  —  Ve(x  -—  a)z, 


which  gives 


az*-\-  6 
^  "";?'  +  1  ' 

a^  _  2(^  -  ^)zdz. 


V2        ) 


V—a-\-bx- 


cx 


V^j-a)z^ 

z*-^l    "' 

substitutions  which  will  render  the  equation  rational. 


(a) 
(*) 


140.  We  have  already  integrated  one  expression  of  tlie 

dx 
form  just  considered,  namely,  -  without  ration- 

Va  +  bx  +  ex* 
alization.     There  is  yet  another  expression  which  admits  of 
being  integrated  by  a  very  simple  transformation,  namely, 

d0  =  — =J?_. 

r  Var*  -}-  br  —  1 

This  is  the  polar  equation  of  the  orbit  of  a  planet  around 
the  sun.     To  integrate  it  d;.'ectly,  we  put 


(*) 

the  equa- 

me  terms 
)ts  of  the 


n 

— 

a)z, 

(a) 

? 

W 

(c) 

1. 

)n 

of  tlie 

it 

ration- 

idmits  of 
imely. 

it  around 


INTEQItATION  OF  IRRATIONAL  DIFFERENTIALS.  237 


1        ,             dx 
X  =  -:     dr  = -„. 

r  X 


We  thus  reduce  the  expression  to 

—  dec 


-\-})X  —  x^ 

Proceeding  as  in  §133,  Case  I.,  we  find  the  value  of  the 
integral  to  bo 


/ 


dr 


—  cos^~  ^^ 


2:r  -  I 


r—  r>nH(~l) 


hr 


r  Var^  -{-  b?'  —  1 

Thus,  e-7r  =  cos(-^> 


2  -  br 


cos' 


r  Via  4-  b' 


r  Via  +  b' 

7t  being  an  arbitrary  constant.     Hence 

2  —  br  ,  „         . 

=  cos  {6  —  7t). 


r  Via  +  b' 

Solving  with  respect  to  r,  we  have,  for  the  polar  equation 
of  the  required  curve, 

2 

**  ~  5  +  |/(4«  +  b')  cos  (^  -  Tt)'  ^"' 

This  can  be  readily  shown  to  represent  an  ellipse.  The 
polar  equation  of  the  ellipse  is,  when  the  major  axis  is  taken 
as  the  base-line  and  the  focus  as  the  pole, 

2 


r  =    ^(^  -  ^')    = 
1  +  «  cos  6^ 


2 


+ 


2e 


«(1  -  e')   '  «(1  -  e') 
Comparing  with  (a),  we  have 

•  a{l  —  e^)  =  J-  =  parameter  of  ellipse  =p; 


cos  6 


2e 
P 


or 


—  jL\ — JH — -_  —  eccentricity  of  ellipse. 


m 


« • 


H 


:li 


t   i     »■, 


238 


TBB  INTEGRAL  CALOVLVS. 


>  ill 


I'-, if'.  • 


Irrational  Binomial  Forms. 

141.  Oeneral  Theory.    An  irrational  binomial  differen- 
tial is  one  in  the  form 

(a-\-hx''Yx'^dx,  (1) 

in  which  m  and  n  are  integers  positive  or  negative,  while  p  is 
fractional. 

To  find  how  and  when  such  a  form  may  be  reduced  to  a 
rational  one,  let  the  fraction  p,  reduced  to  its  lowest  terms,  be 

T 

-;  and  let  us  put 

y  =  (a  +  Ix^y.  (3) 

This  will  give,  when  raised  to  the  rth  power  and  multi- 
plied by  x^dx,  1 

(a  +  Ix'^Yx^dx  —  x'^y^dx,  (3) 

We  readily  find,  from  (2), 

Ix""  =  y»  -^a;  (a) 

_sy'-'dy  ^ 


dx  = 


bnx 


n-l  > 


'K 


y^iTdx—  r~x 
^  bn 


TO  —  n  +  la/r  +  «— 1 


r 


'dy; 


or,  substituting  for  x  its  value  from  (a), 


x^y-dx  = -iJy^-^Y  ""^  y^'^'-'^dy. 


(4) 


This  last  differential  will  be  rational  if -—  is  an  in- 


n 

teger,  which  will  be  the  case  if  — ^t_  ig  an  integer.  We  shall 

call  this  Case  I. 

To  find  another  case  when  the  integral  may  be  rationalized, 
let  us  transform  the  given  differential  (1)  by  dividing  the  bi- 
nomial by  a;"  and  multiplying  the  factor  outside  of  it  by  a;"^, 
which  will  leave  its  value  unchanged.     It  will  then  be 


1  differen- 

(1) 

while  p  is 

luced  to  a 
fc  terms,  be 


(2) 
md  multi- 

I 

(3) 
(a) 


(4) 


1. 

-  IS  an  m- 


We  shall 

tionalized, 
ng  the  bi- 
it  by  a;"^, 
be 


INTEGRATION  OF  IRRATIONAL  DIFFERENTIALS.  239 


(h-\-ax-''yx'^^'*'dXy 


{!') 


which  is  another  differentia^  of  the  same  form  in  which  n  is 
changed  into  —  n  and  m  into  m  +  np.     Hence,  by  Case  I., 

this  form  can  be  made  rational  whenever  —       ^         is  an 


n 


integer;  that  is,  when  — "^ \-p^'&  such. 


n 


We  have,  therefore,  two  cases  of  integrability,  namely: 

w  +  1 


Case  I. :    when 


n 


=  an  integer. 


Case  II. :  when  — — — |- jo  =  an  integer. 
Bemark.     It  will  be  seen  that  all  differentials  of  the  form 

r 

(a  +  hx^Yx^dx  must  belong  to  one  of  these  classes,  because 

— ^ —  IS  an  integer  when  m  is  odd,  and  — ^ h  9-  is  such 

when  m  is  even.     In  this  statement  we  assume  r  to  be  odd, 
because  if  it  is  even  the  original  expression  is  rational. 

142.  If,  in  Case  I.,  the  integer  is  +  1^  that  is,  if  m  +  1 
=  n,  then  the  expression  can  be  integrated  immediately. 
For  (4)  then  becomes 


— «*•+•-! 


hrt 


y 


dy, 


the  integral  of  which,  after  replacing  y  by  its  value  in  (2),  be- 
comes 

/(a  +  J.T-"-.<^.=  (|+^'  +  c.  (5) 

Again,  if  the  integer  in  Case  II.  is  —  1,  we  have 

m-\-l-\-  np  =  —  n, 
or  m  -\-  np  =  —  n  —  1. 

The  expression  (1)  reduced  to  the  form  (!')  will  then  be 
{b  -f  ax-'^yx-^'-^dx  =  —  (5  +  ax-'')^  — d{b  +  aa;""). 


( 


i 


s  ; 


1/ 


■'  i 


m 


m 


240 


THE  INTEGRAL  CALCVLIS. 


I 


which  is  immediately  integrablc,  and  gives  by  simpV  reduc- 
tions 


Ha  +  hx^'Yx  -  («p  +  "  ^  ^)dx  =  c  - 


(tn{2) -^  l)x''^^  '  ^^' 


(C) 


143.  Forms  of  Reduction  of  Irrational  Binomials,  Al- 
though the  integrable  forms  can  be  integrated  by  the  substi- 
tution (2),  it  will,  in  most  cases,  be  more  convenient  to  ap- 
ply a  system  of  transformations  by  which  the  integrals  can  bo 
reduced  to  one  of  the  forms  just  considered.  The  objects  of 
these  transformations  are: 

I.    To  replace  m.  by  m  -f-  n  or  in  —  w; 
II.  To  replace  ;;  by  ;>  -f-  1  or  ;>  —  1. 

144.  Firstly,  to  replace  m  by  m  -f-  n.  Lot  us  write,  for 
brevity, 

which  will  give  dX  =:  bnx^~hlx, 

and  the  given  differential  will  bo 

XPx^'dx, 
which,  again,  is  equal  to 


X 


m  —  n  +  1 


XPdX  = 


X 


m  — n  +  1 


bn  bn{p-{- 1) 

Integrating  by  parts,  we  have 
r  YP  ^1    -  a:"'-"  +  ^J^^+^  _  7n-n  +  l 

J  X  X  ax  -    ^^^^^  ^  ^^        ^^^^^^  ^^^ 


rf(X^+»). 


^-.JXt'^H'^-^dx.  {a) 


Since 


X^  +  ^  =  XP{a  +  bx^)  =  aXP  +  bX^x^ 
the  last  integral  in  the  above  equation  is  the  same  as 

a  fx^x  '^-''dx-\-b  Cx^x'^dxy 

of  which  the  second  integral  is  the  same  as  the  original  one. 
Making  this  substitution  in  («),  and  then  solving  the  equa- 


tiol 


wh| 

de] 
do 


INTEGRATION  OF  IRli A  TTONAL  DIFFKIiENTlALS.  241 


tion  BO  as  to  obtain  tlio  value  of   /  X^x'^^dz,  we  fiud 

/  X'x'^dx  =  y. ; -,-T  -  77^ — ■ — ^-.{  /  X^x'^-^'dx.  {A) 

J  b{ni}-\- m -\-X)      h{)ii)\-m-\-\)J  ^ 

Thus  the  given  integral  is  made  to  depend  npon  another  in 
which  the  exponent  of  x  is  clianged  from  m  to  m  —  n. 

By  reversing  the  equation  we  malfe  the  given  integral 
depend  on  one  in  which  the  exponent  is  increased  by  n.  To 
do  this  we  change  m  into  m  + 1^  all  through  the  equation 
(A),  thus  getting 

j-p  +  i^m  +  i  a{m-\-l) 


f^ 


XPx^^\lx=z 


X^x'^dx. 


l(np-\-m-\~n-{-\)     b{;np-\-m^n-\-iy 
Solving  with  respect  to  the  last  integral,  we  find 

/  X^x'^dx  =  —. —r.- ^-^ — ~  —-- ^  /  .1  Px'^^^'dx.  IB) 

V  a{m  +  1)  rt(?^i  +  1)       *^ 

The  repeated  application  of  {A)  and  (/>')  enables  us  to 
make  the  value  of  the  given  integral  depend  upon  other  in- 
tegrals of  the  same  form,  in  which 

w  is  replaced  by        m-\-n},    m-\-%}i\    etc.; 

or  by  m  ~  n\    m  —  2h;    etc. 

145.  Next,  to  obtain  forms  in  which  p  is  increased  or 
diminished  by  unity,  we  express  the  given  differential  in  the 
form 


X^x^^dx  =  X^d 


X 


m  f-1 


\m.  -f-  h 

Integrating  by  parts  and   substituting  for  dX  its  value 
bnx^~^dx,  we  have 


f 


X^x^dx  = 


m  -f- 1        m  4- 


■^rx^-'x^''+\lx.      (b) 


Now,  we  have 


X 


m  +  n 


hffi^ti 


X'^X 


_  x'^(X  —  a)  _  Xx"^ 


ytfti 


ax^ 
~b* 


I 


■i 


i   ;, 


'! 


!il 


i 


.1  n  * 


242 


THE  INTEORAL  CALCULUS. 


and  therefore,  by  multiplying  by  JL^~^dXf 

Substituting  this  value  in  (d),  and  solving  as  before  with 
respect  to  /  X^x^dx^  we  shall  find 

^^P      -fx^-^x-dx,  (C) 


fX'x^dx  =  --r~-T^  + 


np  -\-  m  -\-  1      np  -\-m  -\- 

in  which  p  is  diminished  by  unity. 

If  we  write  jo  +  1  for  j»  in  this  equation,  the  last  integral 
will  become  the  given  one.  Doing  this,  and  then  solving 
with  respect  to  the  last  integral,  wo  find 

X^x'^dz  = 7 — r-Tv  +        ,      .   il     /  X^^x'^dxAD) 

By  the  repeated  application  of  the  formula  (0)  or  (D)  we 

change 

p  into  jo  —  1,  p  —  2,  p  —  d,  etc., 

or  p  into  p  -{- 1,  p  -{-  2,  p  -\-  3,  etc. 

146.  To  see  the  effect  of  these  transformations,  let  us 
put,  in  the  ciiteria  of  Cases  I.  and  II.,  §  141: 

m-\-l 


I. 


n 


=  i,  an  integer. 


II.  — — — [-p  =  i',  an  integer. 


n 


Then  when  we  apply  formula  (A)  or  (B),  since  we  replace 
mhj  m  —  n  ov  m  -\-  rif^e  have,  for  the  new  integers: 

m  T  w  +  1 


I. 


n 


=  i  T  1. 


II.  "l^i^-i+^  =  z'Tl. 


n 


It  is  also  clear  that  by  (C)  and  (/))  we  change  II.  by  unity. 
Thus,  every  time  we  apply  formulae  (A),  (B),  (C)  or  {D) 
we  change  one  or  both  of  these  integers  by  unity,  so  that  we 
may  bring  them  to  the  values  unity  treated  in  §  142. 


INTEGRATION  OF  IRRATIONAL  DIFFERENTIALS.  243 

147.  Case  of  Failure  in  this  Reduction.  If,  in  an  integral 
of  Case  II.,  i'  is  positive,  we  eannot  change  it  from  zero  to  —  1 

by  the  formula  (-4)  or  ((7),  because,  when  — — \-  p  =  Q, 

we  have 

m  -{•  \  •\-  np  =  0, 

and  the  denominators  in  {A)  and  (C)  then  vanish.  In  this 
case  we  have  to  apply  the  substitution  of  §  141,  without  try- 
ing to  reduce  the  integral  farther. 

EXAMPLES  AND   EXERCISES. 

1.  To  integrate 

We  see  that  if  we  diminish  the  exponent  |  by  unity,  we 
shall  reduce  the  integral  to  a  known  elementary  form  of  §  135J. 
So  we  apply  (C),  putting 

m  =  0;    w  =  2;    p  -=•  \',    a  =  a*;    b  =  ±  1, 

Then  (C)  becomes 

We  therefore  have,  from  §  133, 
f(a'  +  x'Ydx  =  1  I  x(a'  +  x^)^  +  a'  log  ^(a;+(a'+  a:')*)  |  • 

/*(«'  -  x')^dx  =  i  I  x{a'  -  cc')*  +  a'  sin<-«  j  +  ^  |  • 

Deduce  the  following  equations: 

2.  /  (c'  —  x^)  xdx      =h  —  i(c'  —  ic')'. 


36''a;' 


r     dx  ,     (c'  -h  xy 


if 


ill 

if 


il 


iiii 


244 


I.  ^; 


m 


'     ysp    ill 


6.  /-<i^)-./. 


ra^  TNTEGBAL  CALCULUS. 
(1  -  ?/)^ 


h- 


y      "  6/ 

7.  /{a"  -  x'ydx    =  ^xia'  -  x')^  +  K  sin  <"«  -. 
Here  apply  formula  (C);  in  the  following  (A). 

8.  f{l  -  x')'x'dx  =  A  -  g  +  -)(!  -  x'f. 

9.  To  reduce  and  integrate  (1  +  x^)^x^dx. 

Here  m  =  d;  n  =  2;  2^  =  h  m -\- 1  =  4:  =  2n.  We  can  therefore 
reduce  the  form  to  Case  I.  by  a  transformation  of  m  into  m  —  n,  for 
which  we  may  use  either  (a)  or  (-4)  of  §  144.    Using  (a),  we  have 


J\l  +  x')^  x^dx  =  (^+^')'^.'  _  |/^(i  _|.  a.2)?  xdx. 


\ 


The  last  integral  can  be  immediately  found,  and  gives  for  the  required 
integral 

id  +  x')h'  -  ^%{1  +  a^'O*.  (a) 

Using  (A),  we  shoul(     ad 


y*(l  +  ^)^  x^dx  =  (1  -I-  x')^  (-.  -  ^g), 


(6) 


a  form  to  which  (a)  can  be  immediately  reduced. 

The  student  will  remurk  that  the  form  (a)  is  reduced  to  (A)  because 
in  the  former  the  exponent  of  X  is  increased  by  1,  which  often  makes 
the  integration  iuconvcnicnt.  But  when  this  increase  of  p  does  not  in- 
terfere v/ith  the  integratiori,  we  may  use  (a)  more  easily  than  (A). 

10,  To  reduce  and  integrate  (1  +  x^)^x^dx. 
Applying  (^1),  we  iind 

Al  +  x'')^  3^dx  =  ^^+^')^  _  4y ^j  _^  ^,^i  ^3^^ 

A  second  application  repeats  the  form  (J)  above,  thus  giving 

/(I +«';». •*.=(!+.')'(? -I' +4). 

ir.  Reduce  and  integrate  {x   f  x'^)^x^dx,  where  m  is  any 
positive  odd  integer,  and  show  that 


^^^t* 


I  therefore 
n,  —  n,  for 
ive 


e  required 
(a) 

(6) 


INTEGRATION  OF  IRBATIONAL  FUNCTIONS.      245 
f  (I -\- x'fx'^  dx 

"^    ^  W+2       {m  +  3)w  "^   (mH-3)?/i(m-2)       /* 

Remark.  Where  the  student  is  writing  a  series  of  transformations  he 
will  find  it  convenient  to  put  single  symbols  for  the  integral  expressions 
which  repeat  themselves.    Thus: 

J'xH'^dx  =  {l);      fx^x'^-''dx=i2);    etc. 
Thus  the  equations  of  reduction  in  the  present  example  may  be  written 


(1)  = 


Xh 


2^m  — 1 


ni  —  1 


X(3); 


m-\-2       m-^2' 


etc. 


m 
ett-. 


12.  Deduce  the  result 

-  ^  I  «K  +  «^')*  +  a'log  C(x  +  4/^M=^'")  [  . 


J 

'it ' 


i'l 


m 


4)  because 
ten  makes 
les  not  in- 
A). 


.  <  i   1 
.'I 


Hi 


1 1 


m 


71  IS  any 


246 


THE  INTEGRAL  CALCULUS, 


k;i 


1  ■■'. 


i  , '-It 

\ !    -S  i 


Slifili 


m 


CHAPTER  V. 

INTEGRATION  OF  TRANSCENDENT  FUNCTIONS. 

When  the  given  differential  contains  trigonometric  or  other 
transcendent  functions  of  the  variable  more  complex  than  the 
simple  forms  treated  in  Chapter  II.,  no  general  method  of 
reduction  can  be  applied.  Each  case  must  therefore  be 
studied  for  itself. 


148.  To  find  the  integrals 

/  e"^  cos  nxdx    and      /  e^  sin  nxdx. 

Since  we  have 

e'^dx  =  -die"^)  =  d[—], 
m  ^  \m/ 

the  integration  by  parts  of  these  two  expressions  gives 


fi) 


fe^  cos  nxdx  = 
/  e^  sin  nxdx  = 


otnx 


cos  nx 


m 
e^''  sin  nx 


m 


J —  Cq^  gin  nxdxi 
mj 

mJ 


Solving  these  equations  with  respect  to  the  two  integrals 
which  they  contain,  we  find 


e^  cos 


,        e"**(?»  cos  nx-\-  n  sm  nx\ 

nxdx  =  — ^^ a  ,  '     -\ 

m  -\-  7r  ' 

„,   .         ,        e'^^lm,  sin  nx  —  n  cos  nx) 

c"^  sm  nxdx  =  — ^ 5— — r -i 

m  +  n 


(2) 


which  are  the  required  values. 

Remark.  These  integrals  can  aho  be  obtained  by  substi- 
tuting for  the  sine  and  cosine  their  expressions  in  terms  of 
imaginary  exponentials,  namely, 


INTEQBATTVN  OF  TRANSCENDENT  FUNCTIONS.  247 


H 


TIONS. 

c  or  other 
c  than  the 
nethod  of 
jrefore  be 


a) 


ves 
xdx'y 

ixdx, 

5  integrals 


m 


by  substi- 
1  terms  of 


2  cos  nx  =  e"*^  -\-e 
2  sin  nx  =  -(e"**  - 


—  nxi 


9 

,—nxt 


). 


and  then  integrating  according  to  the  method  of  §  134.    The 
student  should  thus  deduce  the  form  (2)  as  an  exercise. 

149.  Integration  of  sin"*  x  cos"  xdx. 
This  form  is  readily  reducible  to  that  of  a  binomial^  and 
that  in  two  ways.     Since  we  have 

cos  xdx  =  tZ'sin  x, 

cos  a;  =  (1  —  sin'  a;)*, 

we  see  that  the  integral  may  be  written  in  the  form 


/< 


n-  1 


(1  —  sin'  x)    2   sin"*  xd'^io.  x; 
or,  putting  y  =  sin  x, 


/< 


n-l 


(1-y')    2   2^-<?y.  (3) 

By  putting  2;  =  cos  a:  we  should  have,  in  the  same  way, 


-/(I  -  ^) 


m  — 1 


'z^dz, 


(i) 


which  is  still  of  the  same  form,  and  is  always  integrable  by 
the  methods  already  developed  in  Chapter  IV. 

If  either  wi  or  71  is  a  positive  odd  integer,  then  by  develop- 
ing the  binomial  in  (3)  or  (4)  by  the  binomial  theorem  we 
shall  reduce  the  expression  to  a  series  containing  only  posi- 
tive or  negative  powers  of  x,  which  is  easily  integrable. 

We  can  also,  in  any  case,  transform  the  integral  so  as  to  in- 
crease or  diminish  either  of  the  exponents  m  and  71  by  steps 
of  two  units  at  a  time,  as  follows: 

sin"*  X  cos"  xdx  =  cos  "  ""  ^  a;  sin"*  xd'  sin  x 


=  C03^~^XC^ — 


m  +  lrf. 


W-f   1   * 


Then,  integrating  by  parts,  we  have 


H*  J 


nil 


ili 


til 


248 


THE  INIEQBAL  CALCULUS. 


A 


sin"*  X  cos"  xdx 


cos  "  ~  ^  .^•  sin  "*  +  ^  a;  .    n  —  1 


—  ,   ^  A -r  Aiii"*  +  *a;  cos""' a;f?a;.  (5) 

m-\-l  m  H-  It/  ^  ' 

Because  mx'^^^x  =  sin"*  a;(l  —  cos"  x)j  the  last  term  is 
equivalent  to 

—  -  /  sin"*  X  cos  "  ~  *  a; — ^  /  sin"*  a;  cos"  xdx. 

m  +  It/  m  +  It/ 

The  last  of  these  factors  is  the  original  integral.     Trans- 
posing the  term  containing  it,  we  find 

{m  +  n)  j  sin*"  x  cos"  xdx  =  sin  "'  +  ^  a;  cos  "  ~  ^  a; 

+  (n  —  1)  y  sin*"  x  cos  "-''  a:c?a;,     (6) 

in  which  the  exponent  of  cos  x  is  diminished  by  2. 

We  may  in  a  similar  way  place  the  given  differential  in  the 
form 


—  sin"*~*  jTf? 


cos  "  +  ^  a; 


and  then,  proceeding  as  before,  we  shall  find 

{m  +  n)  I  sin*"  x  cos"  xdx  —  —  sin  *"  -  ^  a;  cos  "  +  *  a; 

+  (?w  —  1)  /  sin*""^  a;  cos"a;c?a;,     (7) 

thus  diminishing  the  exponent  of  sin  x  by  3. 

By  reversing  these  two  equations  we  get  forms  in  which 
the  exponents  are  increased  by  2.  Writing  n  -\-%  for  n  in 
the  first,  and  m  -\-  %  for  m  in  the  second,  we  find 

(n  -f  1)  /  sin"'  X  cos"  a;f7a;  =  —  sin  "*  +  ^  a;  cos  "  +  *  a; 

+  {m  -\-n-\-Tj  I  sin*"  a;  cos  "  +  ^  xdx\    (8) 

(w  -f- 1)  /  sin"*  X  cos"  xdx  =  sin  "*  +  ^  a;  cos  "  "*"  *  a: 

+  (m  4-  n  +  3)  y  sin  "» +  ^  3.  ^j^gn  ^^^^^     ^qj 


/^ 


INTEGRATION  OF  TRANSCENDENT  FUNCTIONS.  249 


xdx,  (5) 
tenn  is 

xdx. 
Trans- 


xdx,    (6) 


;ial  in  the 


xdx,    (7) 

in  whicli 
3  for  n  in 


xdx\    (8) 


xdx,     (9) 


150.  Special  cases  of  I  sIq"*  a;  cos"  xdx. 

If  wi  is  zero  and  n  is  positive,  we  derive,  from  (6), 


cos"  xdx  — 


sm  X  cos' 


x    .    ?i 


+ /  cos**~^  xdx'f 


,n  — 3 


?i-3 


.-     ,        sm  ic  COS"--' a;  ,  n  — '6   n      „    .     , 
^  a:<w  = ^^  /  COS"-*  xdx\ 


Mio) 


/  COS  " 

J  n 

etc.  etc.  etc. 

The  integral  to  be  found  will  thus  become  that  of  cos  xdx 
when  n  is  odd,  and  that  of  dx,  or  x  itself,  when  n  is  even. 
The  given  integral  is  then  found  by  successive  rubstitution. 

We  find  in  the  same  way,  from  (7), 

m  —  1 


sin*"  xdx  =  — 


cos  a*  sin  *"-^  a; 


-  + 
m  m 

cos  a;  sin  "'"'a;  .  m  — 3 


r^m^'^xdx; 

A ^^r  /  sin'"-*a:f/a;: 

m  —  2t/ 


w-2 
etc.  etc. 


Mil) 


/ 

/  sin  '""^  xdx  = 

etc. 

From  (8)  and  (9)  we  derive  similar  forms  applicable  to  the 
case  of  negative  exponents. 

EXERCISES. 

I.  /  sin'  X  cos'  xdx.  Ans,  \  cos*  x  —  ^  cos'  x. 


2,  I  sin'  X  cos'  xdx, 

/cos'  xdx 

4.  /^sin'  X  tan'  a;c?a;. 
6.  /  e'"  sin  Si/f^y. 

8.  /  e'"  sin  «/  cos  ydy. 
10.  Derive  the  formulae  of  reduction 

/'tan'"  j(?a;  =  - — -r—^ /'tan  •"  + 


^«5.  \  sin'  a;  —  ^  sin'a:. 

.        3  sin'  a;  —  1 

Ans.  — ^ — r-5 . 

3  sm  X 

/cos'  X  , 

7.  A  *  +  «  cos  (a;  +  ^))  c?a;. 
9.  /  G~^  cos'  (y  -f-  «)  dy. 


^  xdx\ 


I  Hi 


litl 


nr 


i 


,i; 


M    , 


ili,         ,  ■    ■■; 


250 

and  hence 


THE  INTEGRAL  CALCULUS. 


/*tan" 


xdx 


tan 


n 


n  —  1  ^  /* 

— -:j /  tan""'  xdx. 


These  equations  may  be  obtained  independently  by  putting  tan"  x 
=  tan » - 2 a;(sec* x—1);  or  they  may  be  derived  from  (5). 


Hence  derive  the  integrals: 

11./  tan*  xdx  =  ^  tan'  x  —  log  c  sec  x. 

12.  /  tan*  xdx  =  ^  tan'  a;  —  tan  x  -{-  x  -{-  h, 

13.  For  all  odd  positive  integral  values  of  71, 
tSLn^~^x      tan**~'a; 


(Cf.  §  127) 


4-  ...  ±  log  6'  sec  X. 


/tan"  xdx  —  ^ 

71—1  71  —  3 

14.  When  71  is  positive,  integral  and  an  even  number, 

/,     „    ,        tan*'~*a;      tan"~'rc    .  .  , 

tan"  xdx  = :; h  •  •  •  ±  tan  x  ±  x. 
71  —  1               71  —  6 

15.  When  the  exponent  is  integral,  odd  and  negative, 

/„     ^             cot"-^a;   ,   cot"-'a;  ,  , 

tan~"a;«a;  = ^ — .  .  .  ±  loff  c-smo;. 
71  —  1              71  —  d  ° 

16.  When  the  exponent  is  integral,  even  and  negative, 

/»^        „     ,  cot**~^a:   ,  cot"~^a; 

/  tan -"  xdx  =  — — | ...  ±  cot  a;  ^  a;. 


71-1 


w-3 


/•  •  5    J             coaxf  .  ,      ,   4    .  ,      ,  4-2\ 
7.     /  sin  xdx  = — -I  sin  x  +  -  sm  ^'  +  oTi  )• 


;./d 


18.     /  sin^  xdx 


cos  a 


-(^sm  .-.  + J  Sin  :.  +  —  sm.rj +^;^. 


^9 


.     /*sin" 


X  cos"  xdx 


^^-/sin" 


22;r/a; 


cos  2.r  sin"~^  2a;   .  71  —  1 


2 


n  +  1 


?J 


-4--Trr—    /  sm" 


-  2 


2a;dr. 


or 


ting  tan"  a: 


Cf.  §  137) 


'  SGC  X, 

iber, 

ban  X  ±  X. 
;ative, 
log  c  sin  X. 
ative, 
cot  x^^  X. 


D- 


+ 


6-dx 


i^'-^^xdx. 


INTEGRATION  OF  TUANISCENDENT  FUNCTIONS.  251 


151.  To  integrate     .^    .  .^^ 


dx 


=  du. 


m  sin  :r  +  71  cos"  x 
Dividing  both  terms  of  the  fraction  by  cos*  Xy  noticing  that 

dx 

=  <^"  tan  a;  and  writing  t  =  tan  x,  we  find 


cos    X 


u  = 


/ 


dt 


The  integral  is  known  to  be^  (§  128) 


(12) 


—  tan<  ^^-^, 
mn  71 


so  that  we  have 


ti 


or 


/; 


dx 


1  m 

i   •  2      ,3  — T—  =  —  tau^~^)  -  tan  x  4-  h,     (13) 
m  sm  X  +  ?i'  cos  x      mn  n  ~    '     ^    ' 


n 


tan  x  =  —  tan  mn{u  —  h). 


152.  Infer/ration  of 


dy 


a  -\-  b  cos  y 

We  reduce  this  form  to  the  preceding  one  by  the  following 
trigonometric  substitution: 

a  =  ^Kcos'i3^  + singly); 
b  cos  y  =  b{cos-  ^y  —  sin'  ^?y); 

by  which  the  expression  reduces  to  the  form 

dm 


'fia 


{a  —  b)  sin'  !?/  +  («  +  b)  cos"  ^y' 
which  is  that  just  integrated,  when  we  put 

x  =  ly', 


(14) 


m  =  Va  —  b; 
7i~Va-\-  b. 


We  therefore  have 


h 


dy 3  _ 


tan< 


:-i)  ^/'L 


-  b 


«+  ^ 


tan  iy  -|-  Ji.    (15) 


?  f 


;  I 


!i 


252 


TUE  INTEGRAL  CALCULUS, 


^u 


153,  If,  in  the  form  of  §151,  ?«'  and  n^  have  opposite 
signs,  or  if  in  §  15:^  we  lir.ve  h  >  a,  imaginary  quantities  will 
enter  into  the  integrals,  although  the  latter  are  real.  If,  in 
the  first  form,  the  denominator  is  m^  sin''  x  —  n^  cos'  x,  we 
shall  have,  instead  of  (ll^i),  the  integral 

r dt_  ^  1   /'    iU    _  1  p_jit  _ 

J  mH^  —  n'^      "ZnJ  mt  —  n      "ZnJ  mt  -\-  n^^       ' 

1    ,      w^  \  n  ,   y 

i^  ■  -  n 


''Zmn 

Hence,  corresponding  to  (Ib^   tvc  L>'vo  thu  result 
(Ix 


/»  (ix 

J  vi^  bin*  x  — 


if  cos'  x 


1  m  tan  .r  +  u  . 

^ —  log  — .  -  - — ' f  h.     (IG) 

2niu        m  tan  x  —  n  ^     ' 


If,  now,  in  §  15:^,  h  >  a,  we  write  (14)  in  the  form 

_  o  r I'i^^ 

^J  (b  —  (()  sin'  hj  —  [a  -\-  b)  cos'  ^y* 

and  instead  of  (15)  we  have  the  result 
dy 


/ 


,  ,         1        ,       Vb—ai'd.\\hi-\-Vb-\-ii   ,^^. 
a+b  cos  y  Vb'-a'         ^b-atan  {y  -  Vb  -^  a 


154.  Intvyrdtion  of  sin  mx  cos  nxclx. 

Every  form  of  this  kind  is  readily  integrated  by  substitut- 
ing for  the  products  of  sines  and  cosines  their  expressions  in 
sines  and  cosines  of  the  sums  and  differences  of  the  angles. 
We  have,  by  Trigonometry, 

sin  mx  cos  nx  =  ^  sin  (m  -\-  n)x  -}-  ^  sin  {m  —  n)x. 

Hence 

/•  .  ,  cos  (ni  4-  n)x      cos  ()n  —  n)x  ,    ^ 

I  sni  mx  cos  nxdx  = 77-^ — ; — { z— r—  +  k 

J  2{ni  +  fi)  2{m  —  71) 

We  find  in  the  same  way 

/sin  (m  4-  7i)x  ,  sin  (m  —  7i)x   ,    , 
cos  mx  cos  7ixax  =       ~-y — ; — r-  H — ~ ^^ — h  «; 
2{)n  -\-  71)            2{m  —  71) 

/sin  im  -\-  n)x  ,  sin  (m.  —  7i)x   ,    , 
sm  nix  sm  7ixdx  = ^> — ,- — r-  -\ — ttt r — h  «. 
2(7)1  -\-  7i)           2(m  —  n) 


INTEGRATION  OF  TRANSCENDENT  FUNCTIONS.  253 


!  opposite 
titles  will 
il.     If,  in 

cos"  X,  we 


13G) 


f  h.     (10) 


m 


±.1  (17) 

substitiit- 
•Gssions  in 
le  angles. 


n)x. 


~  +  h 


%)x 


+  A; 


'f+k. 


155.  Integration  hij  Development  in  Series,  When  the 
given  derived  function  can  be  developed  in  a  convergent 
series,  we  may  find  its  integral  by  integrating  each  term  of 
the  series.  Of  course  the  integral  will  then  be  in  the  form 
of  a  series.  The  development  of  many  known  functions  may 
thus  be  obtained. 

EXAMPLES   AND   EXERCISES. 

I.  We  may  find  /  sin  xdx  as  follows:  We  know  that 

sm  X  —  x      q"!  "  5  f      7  f   '  •  *  *  ^ 
.  • .  /sin  xdx  =  h  -\.  —  __-|-_  —  etc., 

which  we  recognize  as  the  development  of  —  cos  x  with  .  -i 
arbitrary  constant  h  ■\-l  added  to  it. 

Of  course  we  may  find  /  cos  xdx  in  the  same  way. 

dx 


2.  To  integrate 
1 


l  +  o;* 
=  (1  +  a:)-^  =  1  -  a;  +  a:« -a;='+ .  . .  ; 


l-\-x 

/dx         T    ,  x^  ,   x^      X    , 


(a) 


IS 


-^x       "   '   ~      2    '   3        4 

/dx 
'      =  log  (1  +  a;).     Hence  {a)  i 

the  development  of  log  (1  +  ^),  when  we  put  h  =  log  1  =  0. 

The  series  (a)  is  divergent  when  a;  >  1.  In  this  case  we 
may  form  the  development  by  the  binomial  theorem  in  de- 
scending powers  of  x,  thus: 


(^  +  1) 


-^  =  a;-i 


X 


-8 


■\-x~^-x-^^ 


•    .    .    . 


Hence  we  derive,  when  a;  >  1, 

log  (a:  -f  1)  =  log  a;  +--—,  +  5773  -  j::;^  + 


'i 


X 


2x^  '  3a;'      4a;* 


J>54 


TUE  INTEQliAL  CALCULUS. 


ilt 


1 


The  arbitrary  constant  is  zero  because,  when  x  is  infinite, 
Jog  (.c  +  1)  ~  log  -^  is  infinitesimal. 

3.  To  find  / ' — -  =  sin  <~  *>  x  in  a  series. 

J    Vl-x^ 


(i-x')-» =1+^.:'  +  x^ + j;f:^^"+ . . . . 


Hence 


/ 


dx 


vr 


X 


,3 


2-4 


•    (-1)    _     ,1    ^'  I   1'3  a:"*      1-3-5  x' 


2   3    '  2-4  5     '  2-4-6  7 


T 

rp 


The  arbitrary  constant  is  zero  by  the  condition  sin^~*^0  =  0. 
This  series  could  be  used  for  computing  n  by  putting  x  = 


71 


\y  because  \  =  sin  30°  =  sin  -.     But  its  convergence  would 

be  much  slower  than  that  of  some  other  series  which  give  the 
value  of  TT. 


4.  From  the  equation  y - 
rive  the  expansion 


dx 


Vl-{-x' 


log  (re  +  1^1  +  x')   de- 


iog(a;+4/i4-^')=^-^.3-+|;||'-|;|;|.y+...> 

dx 
5.  By  expanding  ,  =  <Z'tan^~*'  x,  derive 

X  ~j~  X 

tan^-»  a;  =  a;  -  K  +  i^;'  -  |a;'  +  .  .  .  . 


Derive: 


-  f 


dx 


/i-\-x—     .jr  +.5n-Q  —  9./^.«■T■o■r•••• 


4/l-^a;*      "  '  "      2   5'  2-4"9       2-4-6   13 
7.  /.-Va;=A-4-a;-|+g-^^-j-^3j4-.... 


ini 


DEFINITE  INTEGRALS. 


265 


infinite  j 


•  w     1    •  •  •  • 

-"0  =  0. 
;ting  X  = 

ce  would 

I  give  the 

fx')   de- 


CHAPTER  VI. 

OF  DEFINITE  INTEGRALS. 

156.  In  the  Differential  Calculus  the  increment  of  a 
variable  has  hcen  defined  as  the  difference  between  two  values 
of  that  variable.  Let  us  then  suppose  u  to  represent  any 
variable  quantity  whatever,  and  let  us  suppose  u  to  pass 
through  the  series  of  values 


^*o^  ^^i»  **a>  ^s>  •  •  •  ^'n« 


Then  we  shall  have 


^u. 

rr 

U, 

— 

«^; 

/lu^ 

z= 

u. 

— 

w.; 

^u^ 

r= 

1^ 

— 

«,; 

• 

• 

• 

• 

• 

• 

~y"    •     •     •    r 


-^W„_l  =  Un  —  Un-x. 

Taking  the  sum  of  all  these  equations,  we  have 

A^l^  +  An,  +  Av^  +  .  .  .  +  J?*n_i  =  ?«„  -  t^„; 

That  is,  the  difference  hetivcen  the  two  extreme  valnes  of  a 
variable  is  equal  to  the  sum  of  all  the  successive  increments 
ly  which  it  passes  from  one  of  these  values  to  the  other. 

The  same  proposition  may  be  shown  graphically  by  sup- 
posing the  variable  to  represent  the  distance  from  the  left- 
hand  end  of  a  line  to  any  point  upon  the  line.  The  differ- 
ence between  the  lengths  Au^  and  Au^  is  evidently  Au^ 
+  Au^  +  .  .  .  +  Au^, 

^  tto  «i  «a         "3  «4  «5 

Since  the  proposition  is  true  how  small  soever  the  incre- 
ments, it  remains  true  when  they  are  infinitesimal. 


! 


256 


THE  INTEGRAL  CALCULU3. 


if. 


i: 


1  '. : 


I  1^ 

U 


If  ! 


Fig.  47 


157.  Differential  of  an  Area,     Lot  P^PP'  bo  any  curvo 

whatever,  aud  lot  us  iuvostigate  the  differential  of  the  area 

swept  over  by  the  ordinate 

XP.     Let  us  suppose  the 

foot    of    the    ordinate  to 

start  from  the  position  X^y 

and  move  to  the  position 

X,      During  this  motion 

XP  swct'ps  over  the  area 

X^PJ^Xy  the  magnitude 

of  which  will  depend  upon 

the  distance  OX,  and  will 

therefore  be  a  function  of  Xy  which  represents  this  distance. 

Let  us  put 

u  =  the  area  swept  over; 

y  E  the  ordinate  XP. 

Then,  if  we  assign  to  x  the  increment  XX',  the  corre- 
sponding increment  of  the  area  will  be  XPP'X'.  Let  us  call 
y'  the  new  ordinate  X'P',  It  is  evident  that  we  may  always 
take  the  increment  XX'  ~  Ax  so  small  that  the  area  XPP'X' 
shall  be  greater  than  yAx  and  less  than  y'Ax  or  vice  versa. 
That  is,  if  y'  >  y,  as  in  the  figure,  we  shall  have 

yAx  <  An  <  y'Ax. 

Now,  when  Ax  approaches  the  limit  zero,  y'  will  approach 
y  as  its  limit,  so  that  the  two  extremes  of  this  inequality  yAx 
and  y'Ax  will  approach  equality.     Hence,  at  the  limit, 

du  =  ydx.  (1) 

That  is,  the  area  u  is  such  a  function  ofx  that  its  differen- 
tial is  ydx,  and  its  derivative  with  respect  to  x  is  y. 
From  this  it  follows  by  integration  that 


u 


=fydx  4- 


h 


(2) 


is  a  general  expression  for  the  value  of  the  area  from  any 
initial  ordinate,  as  X^P^  to  the  terminal  ordinate  XP, 


DEFINITE  INTEGRALS. 


257 


il 


any  curve 
\  the  area 


X  X 


s  distance. 


the  corre- 
Lct  us  call 
nay  always 
I  XPP'X' 

vice  versa. 


approach 

iiality  y^x 
tnitj 

(1) 

s  differen- 


(2) 
from  any 


i» 


158.  The  Conception  of  a  Definite  Integral,  Suppose 
the  area  X^  1\  P  X  ee  u 
to  bo  divided  up  into 
elementary  areas,  as  in 
the  figure.  This  area 
will  then  be  made  up  of 
the  sum  of  the  areas  of 
all  the  elementary  rect- 
angles, plus  that  of  the  oi 
triangles  at  the   top  of  *'»o-  48. 

the  several  rectangles.     That  is,  using  the  notation  of  §  15C, 
we  have 

u  =  y,Ax,  -f-  y,Ax^  +  yjx^  +  .  .  .  -f  ?/„_,^:r„_i  +  T, 

T  being  the  sum  of  the  areas  of  the  triangles;  or,  using  the 
notation  of  sums, 


u 


:    :S  y,Ax,  +  T. 


Now,  let  each  of  the  increments  Axf  become  infinitesimal. 
Then  each  of  the  small  triangles  which  make  up  T  will  be- 
come an  infinitesimal  of  the  second  order,  and  their  sum  T 
will  become  an  infinitesimal  of  the  first  order.     We  may 
therefore  write,  for  the  area  w, 

x-OX  x=OX 

u  =  lim.   2    yAx  =    2    ydx. 

x=OXo  x—OXa 

That  is,  u  is  the  limit  of  the  sum  of  all  the  infinitesimal 
products  ydx,  as  the  foot  )f  the  ordinate  XP  moves  from  X^ 
to  JTby  infinitesimal  steps  each  equal  to  dx. 

Such  a  sum  of  an  infinite  number  of  infinitesimal  products 
is  called  a  definite  integral. 

The  extreme  values  of  the  independent  variable  x,  namely, 
OX^-^x^  and  OXe  x^,  are  called  the  limits  of  integration. 

The  infinitesimal  increments  ydx,  whose  sum  makes  up  the 

definite  integTal,  are  called  its  elements. 
17 


li 


t    ■ 


<; 


258 


THE  INTEGRAL  CALCULUS. 


'II 

I  * 

liij 


^lir 


] 


I 


i^ 


|;:|ii^ 


11^ 


|- 


159.  Fundamental  Theorem.  The  definite  integral 
of  a  continuous  function  is  equal  to  the  difference  bctioeen  the 
vahies  of  the  indefinite  integral  corresponding  to  the  limits  of 
integration. 

To  show  this  let  js  write  t})(x)  for  y,  and  let  us  put,  for  the 
indefinite  integral, 

<p(x)dx  r=  F(x)  -j-  c. 


f 


Now,  as  already  shown,  this  is  a  general  expression  for  the 
area  swept  ovt'.r  by  the  ordinate  y  =  <p{x),  when  counted  from 
any  arbitrary  point  detenained  by  the  constant  c.  It  we 
count  the  area  from  X„P„,  the  area  will  be  zero  when  x  =  x^; 
that  is,  we  must  have 

F{x,)  +  c  =  0, 

which  gives  c  =  ~  F{x^), 

If  we  call  x^  the  value  of  x  at  X,  we  shall  have 

u  =  Area  X,P,PX  =  F(x,)  -f  c  =  Fix,)  -  F(xX     (3) 

which  was  to  be  proved. 

We  therefore  have  a  double  conception  of  a  definite  in- 
tegral, namely: 

(1)  As  a  sum  of  infinitesimal  products; 

(2)  As  the  difference  between  two  values  of  an  indefinite 

integral; 
and  it  will  be  noticec'  that  the  identity  of  these  two  concep- 
tions rests  on  the  theorem  just  enunciated. 

Notation.  The  definite  integral  is  expressed  in  the  same 
form  as  the  indefinite  integral,  except  that  the  limits  of  inte- 
gration are  inserted  after  the  sign  /  above  and  below  the  line; 
thus^ 


(p{x)d'. 


X 


means  the  integral  of  <p{x)dx  taken  between  the  "'imitg  x^  and 
a;,,  the  first  being  the  initial  and  the  second  the  terminal  limit 


te  integral 
hettveen  the 
'le  Uinits  of 

)ut,  for  the 


ion  for  the 

mted  from 

c.     If  we 

len  X  =  x^; 


n^.).  (3) 

efinite  in- 
indefinite 
vo  concep- 

the  same 
its  of  inte- 

wthe  line; 


tii^s  x^  and 
inal  limit 


DEFimTE  INTEGRALS. 


9M 


Example  of  the  Identity  of  the  Two  Conceptions  of  a  Defi- 
nite Integral.  The  double  conception  of  a  definite  integral 
just  reached  is  of  fundamental  i.  portance,  and  may  be 
further  illustrated  analytically.  To  take  the  simplest  possible 
case,  consider  the  definite  integral 


/    adx, 

"IXo 


a  being  a  constant.     By  definition  thig  means  the  sum  of  all 
the  products 

adx  +  itdx  +  adx  -f  .  .  .  , 

as  X  increases  from  x^  to  x^.     The  sum  of  all  the  dx's  must 
be  equal  to  a;,  —  x^  (§  150'      Hence 

a{dx  -\-  dx  -^^  dx  4-  dx  -{-...)=  a{x^  -—  x^. 
But  we  have  for  the  indefinite  integral 


/ 


adx  =  ax; 

and  the  definite  integral  is  therefore,  by  the  theorem, 

ax^  —  ax^    or    a{x^  —  x^), 
as  before. 

160.  Differentiation  of  a  Definite  Integral  with  res^pect  to 

its  Limits. — Because  the  definite  integral  /    ydx  =  7i  means 

the  sum  of  all  the  products  yd.v  as  x  increases  by  infinitesimal 
increments  from  the  lower  limit  x^  to  the  upper  limit  x^,  or 

u  =  y^dx  -j-  y'dx  +  y"dx  +  .  .  .  +  y^'^^dx, 

therefore,  assigning  an  increment  dx^  to  the  terminal  limit 
x^  will  add  the  infinitesimal  increment  y^dx^  to  u  (see  Fig.  48). 
That  is,  we  shall  have 

ntt. 


du 


du  =  y,dx^,    or    -^  =  y,  =  0(a;,). 

In  the  same  Avay,  increasing  the  initial  limit  x^  by  dx^  will 
take  away  from  the  sum  the  infinitesimal  product  y^dx^y  so 


ni 


•  1 


i! 


Si 


:'i 


<  j'sV 


I; 


ill; 


260 


rirfi?  INTEGRAL  CALCULUS. 


that  we  shall  have 


^_  =  -  2/.  =  -  0W. 


(5) 


The  equations  (4)  and  (5)  give  us  the  derivatives  of  the 
definite  integral 

u  =  I     (p{x) .  dx 


with  respect  to  its  limits  x^  and  x^. 

161,  Examples  cmd  Exercises  in  finding  Definite  Inte- 
grals. 

The  fundamental  theorem  gives  the  following  rule  for  form- 
ing definite  integrals: 

1.  Form  the  indefinite  integral. 

2.  Substitute  for  the  variable  loith  respect  to  ivhich  toe  inte- 
grate, firstly,  the  upper  limit  of  integration;  secondly,  the 
lower  limit. 

3.  Subtract  the  second  result  from  the  first.  The  difference 
10 ill  be  the  required  definite  integral. 

1.  f  'xhlx  =  ^x^'  -  ix;\ 

2.  /    xdx  =  l{b*  —  «').       3.     /  xdx  —  ^. 
4.    I    sin  xdx  =  —  cos  ;r  -{-  cos  0  =  2. 

<:.     ^    cos  xdx  =■  sin  ^tt.      6.     /    azdz  =  ^a(a*  —  b*). 
^     Jo  Jh  ^    \  I 

8.     /     cos  ^xdx, 

10.     /    cos'  xdx, 

I  X  mi  xdx,  12.    /  z  Goa  zdz. 

Jo  Jo 


.ifl- 


7.     /     sin  2.?;di:. 
9.     /     sin'  xdx. 


II. 


13. 


/ 


2;'  sin  zdz. 


14. 


/ 


2;'  cos  zdz. 


DEFINITE  INTBOBALS. 


(5) 

ives  of  the 


Unite  Inte- 
e  for  form- 


'ch  we  i?iie- 
:ondly,  the 

e  difference 


t*  -  b'). 


261 


,2ir 


15.    /    2'  COS  2zdz, 

•'•  /  T- 

»^     dx 


19. 


/"     ax 


21.    /      coaxdx. 


23 


£ 


IT 

+  3   dz 


'  J^2  z^-r 

25.     /  (^  —  (i)dx, 

t/n  —  6 
1  -X 


27.    /         (x  —  l)'^a;. 

t/i  +  0; 

pb  nn 

29.     /   ?/(/y  +  /   xdx. 


16.    /    2'  sin  22;c?2;. 
t/o 

/a 
nzHz, 

22.     /      sin  xdx. 

24.    /     (^  —  lydz. 

26.     /         ;yr/y. 

t/a  —a; 


3^ 


/  "^  sin  au^-Ja;. 
:^T,.  Deduce    /       cos  (c  +  y)dx  =  sin  2^. 

34.  Show  that  /'y(:r)r/x=  -  /*/(;?;)f?a;. 

35.  Deduce    /     o'^dy  ~  1. 

36.  Deduce    /     e'^^^ dy  —  ~, 

Jo  a 

37.  Deduce   /     eydy  —  1. 

/»« 

38.  Deduce   /     c  -  v'^ijdy  =  |. 

39.  Deduce   f      ^-  — ,  —  tt. 

»/_ao       1  +  2 


28.     /         (ic  —  «)  (a:  —  c)dx. 

t/a  -  c 
/»+  ^    dz  7t 

32.     A'  cos  (a  4"  ^')^^'* 


^      dz 


40.  Deduce   /  — 

./o    4/1  — 


41.  Deduce 


0    Vl 


/ 


a    \  a 


V'c 


71 


=  TT, 


l\ 


1^' 


^f 


■It 


ip^:^^f 


i  '■ 


'"■: 


i\  j- 

ii 
111 


262 


THB  INTEGRAL  CALCULUS. 


16^.  Failure  of  the  Method  when  the  Function  becomes 
Infini'(.  It:  IS  to  be  noted  that  the  equivalence  of  the  twc 
conceroiono  of  a  definite  integral  does  not  necessarily  hold 
true  auless  the  function  y  or  cf){pc)  is  continuous  and  finite 
between  the  limits  of  integration.  As  an  example  of  the 
failure  of  this  condition, 
consider  the  function 

_        1 

y  ~  {x-  ay* 

the     curve     representing 
which    is    shown    in   the 
margin. 
The  indefinite  integral  is 

—  /  ydx  ■=  c .     ^ 

*J  -^  X  —  a      ^ 


II 


MA 

Fio.  49. 

To  find  the  value  of  this  integral  between  two  such  limits 
as  0  and  k,  k  being  any  quantity  OM  less  than  a,  we  put 
2;  =  0  and  X  =  ky  and  take  the  dilference  as  usual.     Thus 

a  —  k      a       a((i  —  k)'  ^   ' 

Now,  if  we  suppose  k  to  approach  a  as  its  liiait,  so  that 
a  —  k  shall  become  infiuitesimal,  then  the  area  u  will  increase 
without  limit,  as  we  readily  see  from  the  figure  as  well  as  by 
the  formula. 

But  suppose  k  >  a;  for  example,  k  =  2a.  Then  the 
theorem  would  give 

2 


/» •«        _  _  1  _  1  - 

Jq    y      ~      a       a  ~ 


a 


a  negative  finite  quu.<>tity;  whereas,  in  reality,  the  area  is  an 
infinite  '"lUf.ntitv. 

The  theorerr-  fp.'Js  bec-iuse,  when  x  —  a,  y  becomes  infinite, 
80  that  ydx  is  not  ihan  necessarily  an  infinitesimal,  as  is  pre- 
supposed in  the  demonstration. 


II  becomes 
i  the  two 
arily  hold 
and  finite 
>le  of  the 


uch  limits 
a,  we  put 
Thus 

(5) 

it,  so  that 
11  increase 
well  as  by 

Then  the 


area  is  an 

es  infinite, 
,  as  is  pre- 


^^.% 


DEFINITE  INTEQBALa. 


}     *< 


263 


163.  Change  of  Variable  m  Dejiuite  Integrals.  When, 
in  order  to  integrate  an  expression,  wo  introduce  a  new  vari- 
able, we  must  assign  to  the  limits  of  integration  the  values  of 
the  new  variable  which  correspoad  to  the  limiting  values  of 
the  old  one.     Some  examples  will  make  this  clear. 

Ex.  1.  Let  the  definite  integral  be 

^    dx 


I 


fQ    a-\-x 

Proceeding  in  the  usual  way,  we  find  the  indefinite  integral 
to  be  log  (a  -f-  a;),  whence  we  conclude 

/";Ffi  =  '"S2«-log«  =  log2. 

But  suppose  that  we  transformed  the  integral  by  putting 

y  =  a-}-x;      dy  =  dx. 

Since,  at  the  lower  limit,  x  =  0,  we  must  then  have  y  =  a  for 
this  limit,  and  when,  at  the  upper  lim't,  x  =  a,  we  have 
y  =  2a.     Hence  the  transformed  integral  is 

^dt/ 

which  we  find  to  have  the  same  value,  log  2. 
Ex.  2.  ««  =  /  2  sin  x{l  —  cos  x)dx. 
We  may  write  the  indefinite  integral  in  the  form 
/  sin  xdx  +  /  cos  xd{co^  x). 

In  the  first  term  x  is  still  the  independent  variable.  But, 
as  the  second  i?  written,  cos  x  is  the  independent  variable. 
Now,  for 


and  for 


x  =  0,      cos  X  =  l'y 
cos  X  =  0. 


_  ^ 


A/ 


Henc^,  writirg 

^0 


rr 


"^'  '^os  X,  the  value  of  u  is 
in  xdx  -f-  /   ydy 


l-^^-h. 


T^. 


ill 


ii 


264 


fik 


I'i 


i 


I  I' 


:'! 


J 


f 


4^! 


2!ffl2?  INTEGRAL  CALCULUS. 


Remabk.  The  variable  ivith  respect  to  xoliich  the  integra- 
tion is  performed  always  disappears  from  the  definite  integral, 
which  is  a  function  of  the  limits  of  integration,  and  of  any 
quantities  which  may  enter  into  the  differential  expression. 
Hence  we  may  change  the  symbol  of  the  variable  at  pleasure 
without  changing  the  integral.  Thus  whatever  be  the  form 
of  the  function  0,  or  the  original  meaning  of  the  symbols  x 
and  y,  we  shall  always  have 

f  (p(x)dx  =  f  (f>{y)dy  =  f        <p{y  -f  a)dy,  etc. 

t/o  t/a  t/0 

164.  Subdivision  of  a  Definite  Integral,  The  following 
definitions  come  into  use  here: 

1.  An  even  function  of  a:  is  a  function  whose  value  remains 
unchanged  when  x  changes  its  sign. 

2.  An  odd  function  of  r  is  one  which  retains  the  same 
absolute  value  with  the  opposite  sign  when  x  changes  its  sign. 

As  examples:  cos  r  is  an  even,  sin  x  an  odd^  function. 

Any  function  of  x^  is  even;  the  product  of  any  even  func- 
tion into  X  is  odd. 

It  is  evident,  from  the  nature  and  formation  of  a  definite 
integral,  that  if  we  have  a  sum  of  such  integrals, 

V>  y»C  pd  pfl 

I    (/){x)dx  +  /    <p{x)dx  4-  /     <f>{x)dx  +  .  .  .  -f  /     <P(x)dXf 

J  a  Jh  Jc  tig 

m  which  the  upper  limit  of  each  integral  is  the  lower  limit 
Ox  that  next  following,  this  sum  is  equal  to 

•  J' 
/     i){x)dx. 

This  theorem  may  often  be  applied  to  simplify  the  expres- 
sion of  the  integral  in  cases  where  the  values  of  0(a-)  repeat 
themselves. 

Theokem  I.  Tf  <p(x)  is  an  even  function  of  x,  then^  what' 
ever  be  «, 


/+a  pa 

<p{x)dx  =  2  /    c/)(x)dx. 


DEFINITE  INTEOltALS. 


265 


e  Integra- 
e  integral, 
nd  of  any 
xpression. 
t  pleasure 
the  form 
symbols  x 

,  etc. 

1  following 

ae  remains 

the  same 
es  its  sign, 
ction. 
even  func- 

£  a  definite 
j     <f){x)dXy 

9 

ower  limit 


the  exprcs- 
(p{x)  repeat 

theii,  what- 


Because  0(—  a;)  =  <p{x)f  it  follows  that  for  every  negative 
value  of  X  between  —  a  and  0  the  element  of  (f){x)dx  will  be  the 
fiame  as  for  the  corresponding  positive  value  of  x.     Hence  the 

infinitesimal  sums  which  make  up  the  value  of  /    (p{x)clx  will 

t/—  a 

bo  equal  to  those  which  make  up   /    (p{x)dx.     Therefore 

/ja  />0  /»a  /-«« 

/    (f)(x)dx  =  I    (p(x)dx  +  /    (p(x)dx  =  2  /    (J){x)dx, 

J -a  J -a  t/0  t/0 

Theorem  II.  If^(x)  is  an  odd  function  of  x,  then,  what- 
ever be  a, 

^{x)dx  =■  0. 

a 

For  in  this  case  each  element  0(—  x)dx  will  be  the  negative 
of  the  element  (p{x)dx,  and  thus  the  positive  and  negative 
elements  will  cancel  each  other. 

EXERCISES. 


ill 


/»8  /,!/  l\n 

I,  Show  that  /    e-'x^dx  =  /   (log  -J  dz. 


substitute  x  =  log  -. 


?.  Show  that  whatever  be  the  function  0,  we  have 

pin  pin 

I     0(sin  z)dz  =  I     0(cos  xdx). 
As  an  example  of  this  theorem. 


/ 


i^rt  +  5  cos"  X 
a  —  b  cos'*  X 


dx 


i'^rt  -f-  b  sin"  X 
b  sin"  X 


t/o    a  — 


dx. 


The  truth  of  this  theorem  may  be  seen  by  sliowing  that  to  each  cle- 
ment of  the  one  integral  corresponds  an  equal  element  of  tlie  other. 
Draw  two  quadrants;  draw  a  sine  in  one  and  an  e(iual  cosine  in  the  otiier. 
Any  function  (p  of  the  sine  is  equal  to  the  corresponding  function  of  the 
cosine.  We  may  lill  one  quadrant  up  with  sines  and  the  other  with 
cosines  equal  to  those  sines,  and  then  the  two  integrals  will  be  made  up 
of  equal  elements. 


(i 


llii 


1 1 


1! 


i 


I    ^ 


■•; 


;  ^; 
I  :■;:;  : 


M;  '■■ 


Ig.i:, 


Ij  !$ 


266 


T/r£?  INTEGRAL  CALCULUS. 


To  express  this  proof  analytically,  we  replace  «  by  a  new  variable  y 
=  i7t  —  X,  Avhieh  gives  sin  x  —  cos  y;  dx  —  —  dy\  and  tben  we  invert 
the  limits  of  the  trausfiirnicd  integral,  and  change  y  into  a  in  accordance 
Willi  the  remark  of  the  last  article. 

IT 

3.  Show  that  r  /(sin  ?)dx  =  2  r'^f(Em  x)dx, 

4.  Show  that  /    0(sin  a:)  cos  xdx  =  0. 

5.  Show  that  if  0  be  an  odd  function,  then 

/     0(cos  x)dx  =  0. 

6.  Show  that  the  product  of  two  like  functions,  odd  or 
even,  is  an  even  function,  and  that  the  product  of  an  even 
and  an  odd  function  is  an  odd  function. 

7.  Show  that  when  0  is  an  odd  function,  0(0)  =  0. 

165.  Definite  Integrals  through  Integration  ly  Parts,— 
In  the  formula  for  integration  by  parts,  namely, 

/  udv  =  uv  —  I  vdu, 

let  us  apply  the  rule  for  finding  the  definite  integral.     To  ex- 
press the  result,  kt  us  put 

(uv)^  and  (tiv)^,  the  values  of  tiv  for  the  upper  and  lower 
limits  of  integration,  respectively; 

udv  and  /     vdu,  the  values  of  the  two  indefinite  in- 
tegrals for  the  upper  limit,  a;,; 

',  the  values  of  the  integrals  for  the  lower 


Jf  udv  and  /  i 


limit 


f  -■■0' 


We  then  have,  by  the  rule  of  §  161, 


udv 


Jf    udv  =  /    udv  —   /  ^ 

=  (w^)i  -  f  '^^w  -  (uv),  +J* 

fJxa 


I 


w  variable  y 
3n  we  invert 
a  accordance 


)ns,  odd  or 
of  an  even 

=  0. 

)y  Parts. — 


ral.     To  ex- 
r  and  lower 

idefinite  in- 
'or  the  lower 


vdu 


DEFINITE  INTEGUALS. 


267 


In  order  to  assimilate  the  form  of  this  expression  to  that  of 
a  definite  integral,  it  is  common  to  write 

EXAMPLES  AND   EXERCISES. 

1.  We  have  found  the  indefinite  integral 

/  log  xdx  =  xlog  X  —  j  dx. 

If  we  take  this  integral  between  the  limits  x  =  0  and  x  =  l, 
the  term  x  log  x  will  vanish  at  both  limits,  so  that 

(x  log  x)^  -  {x  log  x\  =  0. 

Hence      /    log  xdx  =—  /    dx  =  —  1  -\- 0  =  —  1, 

t/o  t'o 

2.  To  find  the  definite  integral, 

/    sin*"  xdx. 

In  the  equation  (11),  §  150,  the  first  term  of  the  second 
member  vanishes  at  both  the  limits  x  =  0  and  x=  n.     Hence 


/ 


sin"*  xdx  = 


m  —  1   n""    . 
m 


t/o 


^xjx"^-^  xdx. 


Writing  m  —  2  for  wi,  and  repeating  the  process,  we  have 

Jr     sin*"~^irc?2;  = -^  /     sin"*~*a;^a;; 
0                           m  —  2e/o 


f    sin*""-* 
«/o 


"^xdx 


m 
m  —  5 


m 


7  /     sin'^'^xdx: 

—  4t/o 

etc.  etc. 

If  m  is  even,  we  shall  at  length  reach  the  form 

/      dx  =  7t  —  0  =  7t. 
Then,  hy  successive  substitution,  wc  shall  havQ 


vu 


' 


h:, 


■  i 


♦  n 


s^ 


;i 


..*.-. 


f 


^pp 


ii-i:  r^ 

I  i   ; 


li'i    if" 


■( ■  :  ■ ' 

.... 

I 


IM 


'11  ;. 


^68  THE  INTEOltAL  CALCULW. 

r  ^m-xdx  =  0^^  -  1)(^  -  3)(m  -  5)  ...  1  ^ 
t/o  'm(7/i  —  ;ii)(«i  -  4)  .  .  .  Jj 

If  m  is  odd,  tho  last  integral  will  bo  /     sin  xdx  =  +  2,  and 
wo  shall  liavo 

Jo  m{7ii  —  2) (m  —  4)  ...  3 

3.  From  the  oquiition  (6)  of  §  140  wo  have,  by  forming  the 
definite  integral  and  dividing  by  m  +  w, 


/ 


sin"*  X  cos"  xdx 


=( 


sin"'^^  a:cos"~^a;\'' 


+ 


n  — 


I 


m  + 


1    /»" 


sin**  a;  cos**"*  xdx. 


Since   sin  ;r  =  sin  0  =  0,  the  first  term  of  the  second 
member  vanislics  between  tho  limits,  and  wo  have 

Jr     siu"*  x  cos**  xd  ■  =  — ; —  /     sin"*  x  cos"""  xdx. 
0  m  +  71  Jo 

Writing  71  —  2,  and  then  71  —  4,  etc.,  in  place  of  71,  this 
formula  becomes 

/*^  71/  —~~  3         z*"^ 

/     sin"*  X  cos""*  xdx  —  — ; /     sin"*  x  cos"~*  xdx\ 

t/o  711  ■\-7l  —  2,/o  ' 

/     sin"*  X  cos""*  xdx  =  — ; /     sin"*  x  cos""*  xdx: 

t/o  m  +  ?j  —  4t/o  ' 

etc.  etc. 

If  71  is  odd,  the  successive  applications  of  this  substitution 
will  at  length  lead  us  to  the  form 


/ 


sin"*  X  cos  xdx 


771  -|- 1 


(sin"*  +  '  ;r-sin'"  +  ^0)  =  0; 


and  thus,  by  successive  substitution, 
tegrals  to  bo  zero, 


we  shall  find  all  the  in- 


A 


'  <! 


.  Tt, 


■\-  2,  and 


rming  the 


8""''  xdx, 
lie  second 

'  xdx. 

of  n,  this 

s**~*  xdx; 
s""*  xdx; 

ibstitution 

^  0)  =  0; 
all  the  iu- 


DEFINITE  INTEGIIALS.  209 

If  n  is  oven,  we  shall  bo  led  to  tlie  form 

/     sin*"  xdx, 

which  we  have  just  integrated.    Then,  by  successive  substi- 
tution, we  find 


/^^ 


u^-  X  cos"  X 


(n  -  l)(n  -  3)  ...  1 


^/    '' 


sin"*  xdx. 


(m  +  ?/)(w  +  ^,  —  sj)  .  .  .  (in  + 

4.  To  find   n--J-l-—-, 
Jo    («'  +  x^Y 

We  transform  the  differential  thus: 

/dx        _  1_      n^x"  +  «"  —  :?;')  dx 

Integrating  the  last  term  by  parts,  we  have 

r     x\lx       _\    r       2xdx      _\  r  d-jx'-^n') 
J  {x'  +  d'Y      2  J  ^\d'  +  x'Y  ~2J  ^  {x'  +  ay 

=  ir^d -1 - 1- ^ 

2J    ""'{x^^dy^-^-     2{)i-\Y{x'-\.dY-'' 


n 


■         1         /*         dx 

Substituting  this  value  of  the  last  term  in  (r/),  we  have 


/ 


dx 


X 


{x^  +  ay    ^A'^^  ~  1)  (;*■'  + «') "  ~ ' 


^aA         2(n-l)W   (x'~^a')^'-'' 


2{ti-l)J^   {x'~\-a') 

Passing  now  to  the  limits,  we  see  that  the  first  term  of  tlie 
second  member  vanishes  both  for  re  =  0  and  for  x  =  (x>.  We 
also  have 

. 1 2)1  —  3 

2{?i  -  1)  ~  2(u  -  1)' 


)    i 


! 


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270 


THE  INTEGRAL  CALCULUS. 


Hence  we  have  the  formula  of  reduction 

dx  2n  —  3      /»*         dx 


/■'^       ax        _    zn  —  6      ff" 
Jo    {x'  +  a'Y  "  2(ir^)7i'Jo    {oF 


+  xY-''       ^^^ 
We  can  thus  diminish  the  exponent  by  successive  steps 
until  it  reaches  2.     The  formula  (b)  will  then  give 

dx  1     /*'*'     dx  n 


Jq    {x'  +  ay  ~  2^e/o    ^ 


+  ^^ 


Then,  by  successive  substitution  in  the  form  (/>),  we  shall 

have 

n"^       dx        _  {2n  -  S){2n  -  5)  ...  1  _;r 

Jo    {x'-{-aY~{2>i-2){2n-^)T772'2a' 


^v     {c) 


If  in  (c)  we  suppose  a  =  1,  and  write  the  second  member 
in  reverse  order,  we  have 

/»« dx 1-3-5  .  .  .  {2n  —  3)  tt 

Jo    (l  +  a:'')»~2-4-6.  .  .  (2w-2)"2* 

»*    x'^dx 


5.  To  find  /  —  = 

Jo    Vl-x' 


y 


m* 


Vl-x- 

Let  us  apply  to  the  indefinite  integral  the  formula  (A), 
144.     We  have  in  this  case 

a  =  1;    J  =  —  ij    n  =  2;    j»  =  —  ^. 
The  f^TTiula  then  becomes 


/- 


x^dx 


X' 


.-i|/l_ 


m  —  1  Px^'hlx 


VI  -x' 
In  the  same  way 


m 


X'  ,  m  —  1  /*  u.       ^v..    ,  . 


7n 


Vl^ 


X 


A 


m  — 8 


dx 


X 


m 


-Wl-x 


Vl-X" 


in- 


—  a;'      m.  -  3    /*jk 
2   "    ~^  m  —  2*/  ^ 


m  — 4 


dx 


V\-x' 


Continuing  the  process,  we  shall  reduce  the  exponent  of  x 
to  1  if  m  is  odd,  or  to  0  if  m  is  even.     Then  we  shall  have 

/xdx  /.         ,v.  f*      dx 

— =  =  —  (1  -  xy    or     /  -—= 


~  =  sin<~*\T.     % 


Vl  -X- 
Taking  the  several  integrals  between  the  limits  0  and  1,  we 


DEFINITE  INTEGRALS. 


271 


sive  steps 


,  we  shall 


in- 


v     (0) 


d  member 


mula  (A), 


~hlx 


l-X' 


m-4 


•  («) 


dx 

onent  of  x 
all  have 

i<-»>.r.     {h) 

)  and  1,  we 


note  that  in  (a)  the  first  term  of  the  second  member  vanishes 
at  both  limits,  while  {b)  gixres 


J/»^      xdx      _  1 .      p^ 
0  i/r^r^~  '  J 


dx 


Vi 


Vin 


Vl  —  x-  1/      r  1  —  a; 

We  thus  have,  by  successive  substitution, 
_  r^x^^'^dx  _  27i(2u  -  2)  (2?i  -  4). 


1 


2 


|/lZr^'      (2w+l)(2>i-l)(2w-3) 


_  p^dx_  _  {2n-l){27i-3){2n-5) 
^^"^  -Jo   |/fZ:"^«  ~  2n{27i  -  2)  {271  -  4).  .' 


2 


.  .3' 

.   .   1    7t 


y  (c) 


t     t     /i     lil 


Let  us  now  consider  the  limit  toward  which  the  ratio  of 
two  values  of  y^  approaches  as  m  increases  to  infinity.  We 
find,  from  («), 

Vm      _  m  —  1 

ym-2  fn    ' 

a  ratio  of  which  unity  is  the  limit. 
Next  we  find,  by  taking  the  quotient  of  the  equations  (r), 

Tt_ I2-4-6  .  .  .  (27i  —  2)'27i}'         y^ 

2  -  I3-5-7T.  .  (2/i  -  l)r(2^Tl)'2^* 

Since,  when  n  becomes  infinite,  the  ratio  y^n  *  .Van  •  i  ap- 
proaches unity  as  its  limit,  we  conclude  that  \7t  may  be  ex- 
pressed in  the  form  of  an  infinite  product,  thus: 

2  ~3*3'5*5'7'7-9*9'ii*  '  *       t7ijimt2im. 

This  is  a  celebrated  expression  for  tt,  known  as  Wallis's 
formula.  It  cannot  practically  be  used  for  computing  7t, 
owing  to  the  great  number  of  factors  which  would  have  to 
be  included. 


il 


!. 


A 


';\ 


272 


TUia  INTEGRAL  CALCULW. 


CHAPTER  VII. 

SUCCESSIVE   INTEGRATION. 

166.  Differentiation  under  the  Sign  of  Integration.    Let 
us  have  an  indefinite  intesral  of  the  form 


u  =  I  (p((x,  x)dx  =  F(a,  x). 


(1) 


a  being  any  quantity  whatever  independent  of  x.     It  is  evi- 
dent that  u  will  in  general  be  a  function  of  a.     We  have 
now  to  find  the  differential  of  u  with  respect  to  or. 
The  differentiation  of  (1)  gives 

^  =  0(«,  ^); 

dhi  _  d(f){a,  x) 
dadx 


T,  d  u        rv  du 

Because  -^ — 7-  =  Da-^r- 

dadx  dx 


da 

Dxi-i  we  have,  when  we  consider 
da 


du 

-T—  as  a  function  of  x  (cf.  §  51), 

\da  I       dxda  aa 

Then,  by  integrating  with  respect  to  x, 

£  =/^fe'..,  (.) 

in  which  the  second  member  is  the  same  as  (1),  except  that 
0(0-,  x)  is  replaced  by  its  derivative  with  respect  to  a.  Hence 
we  have  the  theorem: 

The  derivative  of  an  integral  with  respect  to  any  quantity 
which  enters  into  it  is  expressed  by  differ entiati7tg  with  re- 
spect to  that  quantity  under  the  sign  of  integration. 


SUCCESSIVE!  INTEGRATION. 


273 


n.    Let 


(1) 

t  is  evi- 
^e  have 


consider 


(2) 

apt  that 
Hence 

\uantity 
with  re- 


16 1.  This  theorem  being  proved  for  an  indefinite  inte- 
gral, we  have  to  inquire  whether  it  can  be  applied  to  a  definite 
integral.  If  we  take  the  integral  (1)  between  the  limits  a;, 
and  a;,,  and  put  u^  and  w,  for  the  corresponding  values  of  u, 
we  have,  for  the  definite  integral, 

r^<p{a,  x)dx  =  F{a,  x^)  —  F(a,  «„)  =  w,  —  w,  =  u^\ 

Then,  by  differentiation, 

du^"^  _  dF(a,  x^  _  dF(a,  x^) 
dot   ~       da  da 

Comparing  (1)  and  (2),  we  have 

rd<t>{a,x)^^  ^  dF{a,x)^ 
J        da  da     * 

whence,  if  x^  and  x^  are  not  functions  of  a, 


(3) 


d^{a,x)^^  ^  dF{a,  x,)  _  dF{a,  x,) 
da  da  da 


(*) 


Hence  from  (3)  we  have  the  general  theorem 
Da  I     (p(a,  x)dx  =  /     Da(p{a,  x)dx. 

That  is,  the  symbols  of  differentiation  and  integration  with 
respect  to  two  independent  quantities  may  he  interchanged  in 
a  definite  inicgralj  provided  that  the  limits  of  integration  are 
not  functions  of  the  quantity  with  respect  to  which  we  differ- 
entiate. 

If  the  limits  x^  and  x^  are  functions  of  a,  we  have,  for  the 
total  derivative  of  u„'  with  respect  to  a  (§  41), 

du'       fdu^^ 


da        \  da  ) 
By  §  160  we  have 


du^^  dx^       du^^  dx^ 
dx^   da       dx^  da' 


dii^ 


dx^ 


-  <t>(oty  X,), 


I  > 


18 


274 


THE  INTEGRAL  CALCULUS. 


Thus  from  (3)  and  (4)  we  have 
du 


'0 

da 


'        /»*»  d<t){(Xy  x) ,     ,     ,,        .dx^        ..        .dx„    ,^. 


This  formula  is  subject  to  the  same  restriction  as  the 
theorem  for  the  value  of  a  definite  integral;  that  is,  (fi{a,  x) 
and  its  derivative  with  respect  to  a  must  he  finite  and  con- 
tinuous for  all  values  of  x  betvjeen  the  limits  of  integration. 

If  this  condition  is  not  fulfilled,  (5)  may  fail. 


EXERCISES. 


m\ 


Differentiate: 


1.  / — ; —  with  respect  to  a,         Ans,  —  C-. — ; — r-.. 
J  x-\-a  ^  J  {x-[-  ay 

2.  jix  -\-  aydx  with  respect  to  a,  Ans.  n  l(x-\-aY-^dx, 

3.  l{x^-{-xyYdxmi\ii[es^Qciioy,    Ans,  2  l(x'-[-x^y)dx, 

4.  /  x^dx  with  respect  to  a,  Ans,  a', 
t/o 

5.  /    x'^dx  with  respect  to  a,  Ans,  8ar'. 

6.  /    a;"Ja;  with  respect  to  or.  -4ws.  =a"(2a' "  +  *—!). 

And  show  that  we  have  the  same  results  in  the  first  three 
cases  whether  we  integrate  the  differential  with  respect  to  a 
or  y,  or  differentiate  the  integral. 

168.  The  preceding  method  enables  us  to  find  many 
integrals,  indefinite  and  definite,  by  differentiating  known 
integrals  with  respect  to  constants  which  enter  into  them. 
Thus,  by  differentiating  with  respect  to  a  the  integral 


/ 


e'^dx  =  -e«*  +  c, 


SUCCESSIVE  INTEGRATION. 


275 


as  the 
(f)(ay  x) 

ind  con- 

'ation. 


dx 

aY~^dx, 
\-x^y)dx. 


«  +  i_- 


!)• 


rst  three 
Dect  to  a 


id  many 
l  known 
bo  them. 


we  find,  after  adding  the  constants  of  integration, 

fx'^dx  =  fe'  -  ?f  +  ^.)e-  +  c; 
«/  \a       a        at       ' 


etc. 


etc. 


which  leads  to  the  same  results  as  integration  by  parts,  and 
is  shorter. 

109.  The  following  is  an  instructive  application  of  this 
and  other  principles.    We  shall  hereafter  show  that 

00 


-  «'» 


From  this  it  is  required  to  find  the  value  of  C 
If  we  put 

whence 

the  corresponding  indefinite  integral  will  be 

Now,  when  y  —  ±  oo ,  we  have  also  x=  ±.  <x>.    Hence 


'jy. 


,        dx 
^       a 


r^"  e--'^  dv =1  r 

«/-oo  '^      aj- 


e       dx  = . 

<»  a 


By  differentiating  with  respect  to  a,  and  simple  reductions, 
we  find 


and  from  this. 


""yV--»V,  =  ^; 


00 


2a' 


-j-  00 


y 


♦.-"'J/'././  — 


dy^T 


3  Vn 


00 


etc. 


4    a" 
etc. 


-' « 


Hi 


J 


il^ 


\<\m' 


II 


276 


TffE  INTEGRAL  CALCULUS. 


1     . 

-  sin  ax, 
a 


EXERCISES. 

1.  By  differentiating  the  integrals 
/  cos  axdx 

/sin  axdx  = cos  ax, 
a 

twice  with  respect  to  a,  prove  the  formulas 

I  x*  cos  axdx  =1 -A  sin  ax  -\ — 5  cos  ax*, 

/  a;  sm  aa:aa;  =  -, 1  cos  ax  A — -.  sm  ax, 

J  W      aJ  'a' 

Thence  show  that  we  have 

Jy'  cos  ydy  =  (y'  -  2)  sin  y-{-2y  cos  y; 
Jy^  sin  ydy  =  (2  -  y')  cos  ?/  +  2«/  sin  y. 

2.  Prove  the  formulae: 

/o                 1                            /»o                      1 
e^'^dx     =-;  (^)  /    a;e"*Ja;  = ^; 

x'e'^dx  =  -,;  (^)  /   x^'e^dx  =  (-  1)"-^. 

00  a  t/—  00  fit 

3.  Show  that  the  preceding  formulae  are  true  only  when  a 
is  positive,  and  find  the  follov/ing  corresponding  forms  when 
a  has  the  negative  sign: 


/  e~°^dx     =  — ;  /  xe'^^'^dx   =  — „; 

Jo  (I  «/o  a 

r  x'e-'"'dx=  -A  C  x'e-'^dx  =  ~  ;  etc. 

Jo  a'  Jo  (I*  ' 


4.  By  differentiating  the  form  of  §  132,  namely, 


dx 


J  \a'  -  xy 
with  respect  to  a,  show  that 


/a. 


dx 


=  sin(-i> 


X 


X 

a 


{a'  -  xy      a'{n'  -  xY 


SUCCESSIVE  INTEOiiAriON. 


277 


170.  Dotible  Integrals.  The  procoding  results  may  ho 
Bummed  up  and  proved  thus:  Let  us  have  an  integral  of  the 
form 

u  =  J(}>(x,  y)dx,  (1) 

and  let  us  consider  the  integral 

J^iidy    or     f^  f<p(x,  y)dxj(hj, 
which,  for  brevity,  is  written  without  brackets,  thus: 

/  /  0(^*  y)dxdy. 

This  expression  is  called  a  double  i     r/ral. 

Theoeem.  The  value  of  an  indefinit"-  double  integral  re- 
mai.is  unchanged  when  ive  change  the  order  of  the  integra- 
tions, promded  that  we  assign  suitable  values  to  the  arbitrary 
constants  of  integration* 

Let  us  put 

u  retaining  the  value  (1).    The  theorem  asserts  that 

/  udy  =  /  vdx. 

Call  these  two  quantities  U  and  F,  respectively.  We  then 
have,  by  differentiation, 

dU  d'U      du       ^,      . 

^  =  ^'  d^y^d^^'^^^'^yy^ 


dV 
d^='> 


=  777  =  ^(^'  y)' 


dydx      dif 
Therefore,  because  of  the  interchangeability  of  differentiations. 


d. 


dU 

dx 


,  dV 

dx 

~dy" 
Then,  by  integration  with  respect  to  y, 

dU  _dV 


ax 


dx 


278 


TUK  INTEQIUL  CALCULUS. 


n 


M 


!! 


and,  by  intcgrution  witli  respect  to  x, 

U=  V-'rcx-\-  c\ 
Putting  c  =  0  and  c'  =  0,  we  have  U  =  F,  as  was  to  bo 
proved. 

171.  By  the  process  of  successive  integration  thus  indi- 
cated we  obtain  the  vahie  of  a  function  of  two  variables  when 
its  second  derivative  is  given.  The  problem  is,  hawng  an 
equation  of  the  form 


dxdy 


=  0(^,  y)y 


(2) 


where  0(;f,  y)  is  supposed  to  be  given,  to  find  ?*,  as  a  func- 
tion of  X  and  y.  This  we  do  by  integrating  first  with  respect 
to  one  of  the  variables,  say  x,  which  will  give  us  the  value 

of  -J—,  because  the  first  member  of  (2)  is  D^-i-.    Then  we  in- 
dy*  ^  '        "dy 

tegrate  with  respect  to  y,  and  thus  get  w. 

As  an  example,  let  us  take  the  equation 


d'u 


=  W 


or    rt.-T-  =  xirdx^ 
dy         ^ 


(3) 


dxdy 

Integrating  with  respect  to  x,  we  have 

du       1    ,  ,  ,  y 
^  =  2*3'+''' 

h  being  a  quantity  independent  of  x,  which  wo  have  common- 
ly called  an  arbitrary  constant.  But,  in  accordance  with  a 
principle  already  laid  down  (§118),  this  so-called  constant 
may  be  any  quantity  independent  of  x,  and  therefore  any 
function  we  please  to  take  of  y. 

Next,  iutegi'ating  (3)  with  respect  to  ?/,  and  putting 

Y=fhdy, 

we  find  ti  =  \xY  +  F  +  X, 

in  which  X  is  any  quantity  independent  of  y,  and  so  may  be 
an  arbitrary  function  of  x.  Moreover,  since  h  is  an  entirely 
arbitrary  function  of  y,  so  is  Y  itself. 


8UCGE88I VE  INTEGRA  TION. 


279 


to  bo 

s  indi- 
8  when 
ing  an 

a  func- 
respect 
e  yalue 

L  we  in- 


(3) 

>mmon- 

with  a 

onstant 

)re  any 


may  be 
entirely 


The  student  iihould  now  prove  this  equation  by  differenti- 
ating with  respect  to  x  and  y  in  succession. 

173.  Triple  and  Multiple  Integrals.  The  principles  just 
developed  may  be  extended  to  the  case  of  integrals  involving 
three  or  more  independent  variables.     The  expression 

(f){x,  y,  z)dxdydz 


fff^ 


means  the  result  obtained  by  integrating  0(;r,  ;/,  z)  with  re- 
spect to  Xf  then  that  result  with  respect  to  y,  and  finally  that 
result  with  respect  to  z.  The  final  result  is  called  a  triple 
integral. 

If  we  call  F{x,  y,  z)  the  final  integral  to  be  obtained,  we 

have, 

d'P{x,  y,  z)        ..  . 

dxdydz     =  ^('^'  y^  ^)> 

and  the  problem  is  to  find  F(x,  y,  z)  from  this  equation  when 

(p{Xf  y,  z)  is  given. 

Now,  I  say  that  to  any  integral  obtained  from  this  equation 

we  may  add,  as  arbitrary  constants,  three  quantities:  the  one 

an  arbitrary  function  of  y  and  z;  the  second  an  arbitrary 

function  of  z  and  x;  the  third  an  arbitrary  function  of  x  and  y. 

For,  let  us  represent  any  three  such  functions  by  the  symbols 

[y,  z],     [z,  a;],     [x,  y], 

and  let  us  find  the  third  derivative  of 

F{x,  y,  z)  4-  [y,  z]  +  [z,  x]  +  [x,  y]  =  u 
with  respect  to  x,  y  and  z.     Differentiating  with  respect  to 
X,  y  and  z  in  succession,  we  obtain 


dx 


du  _  dF{x,  y,  z)  d[z,  x]      d[x,y]^ 

dx             dx  dx             ^"^     ■ 

d^u  _  d^'Fi^x,  y,  z)  d'[x,y']^ 

dxdy           dxdy  dxdy  ' 


d\t 


d*F{xyyyz)^ 
dxdydz    * 


dxdydz 

an  equation  from  which  the  three  arbitrary  functions  have 
entirely  disappeared. 


!  in 


JH: 


280 


THE  INTRORAL  CALCULUS. 


i 


It  is  to  be  romarkcd  that  ono  or  both  of  tho  variables  may 
disappear  from  any  of  these  arbitrary  functions  without  chang- 
ing their  character.  The  arbitrary  function  of  y  and  z,  being 
any  quantity  wliatever  that  does  not  contain  x,  may  or  may 
not  contain  y  or  z,  and  so  with  tho  others. 

As  an  example,  let  it  be  required  to  find 

u  •=  I  I  I  {x  —  a)(y  ^  b){z  —  c)(lxdydz. 

Integrating  with  respect  to  z^  and  omitting  the  arbitrary 
function,  we  have 

/  /  i(^  —  «)^y  —  h)(z  —  cydxdy. 

Then  integrating  with  respect  to  y, 

tl  =f'^^''  - ")  ^y  -  *)'  <'  -  "■>'' 

which  gives,  by  integrating  with  respect  to  x,  and  adding  the 
arbitrary  functions, 

u  =  i{x-  ay(y  -  l)\z  -  c)'  +  [y,  z\  +  \z,  x\  +  [a;,  y\. 

The  same  principle  may  be  extended  to  integrals  with  re- 
spect to  any  number  of  variables,  or  to  multiple  integrals. 

The  method  may  also  be  applied  to  tho  determination  of  a 
function  of  a  single  variable  when  the  derivative  of  the  func- 
tion of  any  order  is  given. 


EXERCISES. 


I.     r  j-^dxdy,  2.     r  r(x  —  a)(y—lYdxdy, 

3.    /    /    jxy^z^dxdydz.  4.    j   j    j^dxdydz. 

5.  fff{^  -  (^Y(y  -  *)(^  -  cydxdydz. 

6.  ff{x  "  aydx\  7.  fff{^  +  ^0'^^'- 

Ans.  (6).  ^g(.r  —  «)*  -\-  Cx  -\-  C,  Cand  C  being  arbitrary 
constants. 


8UCGESSI VE  INTEUHA  TION. 


281 


lo8  may 

chang- 

z,  being 

or  may 


rbitrary 


ling  oho 

mi\\  re- 
egrals. 

ion  of  a 
le  f  unc- 


lydxdy. 
iz. 


dz\ 
arbitrary 


173.  Defuiile  Double  Integrals,     Lot  U  bo  any  function 

ppoBing  y 


ot  X  and 


»y 


Integration  with  respect  to  x 
constant,  we  may  form  a  deliuite  integral 


sill 


_ 


Udx  =  U\ 


From  what  has  been  shown  in  §  103,  Rem.,  U'  will  be  a 
function  of  y,  x^  and  .r,.  VV^o  may  thiuefore  form  a  seeonii 
definite  integral  by  integrating  IJ'dy  between  two  limits  //„ 
and  y^.    Thus  wo  lind  an  expression 


f    VUly  =  /     /     Udxdy, 


which  is  a  definite  double  integral. 

The  limii/S  x„  and  x^  of  the  first  integration  may  be  con- 
stants, or  they  may  be  functions  of  ;/. 

If  they  are  constants,  the  two  integrations  will  be  inter- 
changeable, as  shown  for  indefinite  double  integrals. 

If  they  are  functions  of  y  they  are  not  interchangeable,  un- 
less we  make  suitable  changes  in  the  limits. 

174.  Definite  Triple  and  Miiltiplc  Integrals.  A  definite 
integral  of  any  order  may  be  formed  on  the  plan  just  described. 
For  example,  in  the  definite  triple  integral 


'      /      /     ^\^y  Vf  z)dxdydz 


the  limits  x^  and  x^  of  the  first  integration  may  be  functions 
of  y  and  z;  while  y^  and  y,  may  be  functions  of  z.  But  z^  and 
z^  will  be  constants. 

So,  in  any  multiple  integral,  the  limits  of  the  first  integra- 
tion may  be  constants,  or  they  may  be  functions  of  any  or  all 
the  other  variables.  And  each  succeeding  pair  of  limits  may 
be  functions  of  the  variable  which  still  remain,  but  cannot 
be  functions  of  those  with  respect  to  which  we  have  already 
integrated. 


I 
i 


283 


THE  INTEGRAL  CALCULUS. 


EXAMPLES  AND   EXERCISES. 


t 


I'.r 


1.  Find  the  values  of 

/     /   xy^dxdy    and      /     /    X'fdxdy. 

It  will  bo  seen  that  in  the  first  form  the  limits  of  x  are 
con,5tants,  and  in  the  second,  functions  of  y. 

First  integrating  with  respect  to  x^  we  have  for  the  indefi- 
nite integral 

fxy^dx  =  ^xY, 

and  for  the  two  definite  integrals 

Ixy'dx^  ^ay, 

r     xy^dx  =  ^y\ 
y 

Then,  integrating  these  two  functions  with  respect  to  y, 
we  have 

Jo  Jy  ^y'^^^^y  =  Vo  ^  ^^  ^  ^*  • 

Let  us  now  see  the  effect  of  reversing  the  order  of  the  in- 
tegrations.    First  integrating  with  respect  tc  y,  we  have 

/   xy^dy  =  ^xb'  =  U. 
Then  integrating  with  respect  to  x,  we  have 

/  Udx  =  /     /   xy  dydx  =  ^a''b*, 

the  same  result  as  when  we  integraicd  in  the  reverse  order 
between  the  same  constant  limits. 

2.  Deduce  /  ^  /   cos  (re  -f  l/W^^^y  =  —  Ji 


8  UCCESSI VE  IN  TEG  It  A  TION. 


283 


f  X  are 
I  indefi- 


jt  to  y, 


the  in- 
lave 


se  order 


3.  Deduce  /     /   cos  {x  —  y)dxdy       =  -|-  4. 

4.  Deduce  /     /    {x  —  (i)(y  —  b)dxdy  =  ia^b^. 

5.  Deduce /*"/*\;i;  -  a){y  -  b)d>dy  =  \(2ab-d')(2ab-b'), 

6.  Deduce  /  "  /      ^ {x—a){y  —  b)dxdy  =  iCb  —  \aW—  \a\ 

lis.  Product  of  Two  Definite  Integrals. 

Theorem,,     The  product    of  the    two  definite  integrals 

f'^'Xdx  and    f^Ydy    is    equal    to    the    double    integral 

Jf*yi  P'^^xYdxdy,  provided  that  neither   integral  contains 
Wn     *JXa 


the  variable  of  the  other. 


For,  by  hypotliesis,  the  integral    /     Sdx=  f/"  does  not  con- 

C/*X/Q 

tain  y.     Therefore 


U  f    Ydy  =  f    UYdy  =  /      /    XYdxdy, 
as  was  to  be  proved. 

176.  The  Definite  Integral  f       e~  ^"^  dx.     This  integral, 

«/  —   CO 

which  v/e  have  already  mentioned,  is  a  fundamental  one  in 
tlie  method  of  least  squares,  and  may  be  obtained  by  the  ap- 
plication of  the  preceding  theorem.     Let  us  put 

/,=.  r^^e-^\lx  =  ^  r^^e-'\lx  =  ^  r'^^e-'\ly.{%U^) 
J-  00  t/o  t/o 


.  +  00      ^4-co  _ 


Then,  by  the  theorem, 

,/o  t/o  t/o        t/o 

Let  us  now  substitute  for  y  a  new  variable  t,  determined 

by  the  condition 

y=ztx. 


'1 


If 


284 


THE  IJSTEQRAL  CALCULUS, 


Since,  in  integrating  with  respect  to  y,  we  suppose  x  con- 
stant, we  must  now 


put 


dy  =  xdt. 


Also,  since  t  is  infinite  when  y  is  infinite,  and  zero  when  y  is 
zero,  the  limits  of  integration  for  t  are  also  zero  and  infinity. 
Thus  we  have 

t/0  t/0 

Since  the  limits  are  constants,  the  order  of  integration  is 
indifferent.  Let  us  then  first  integrate  with  respect  to  x. 
Since 

xdx  =  i^d'x"  =  2n^i^\^'  (^  +  ^'')^^ 

the  integral  with  respect  to  x  is 


2(1  +  nJo 


-  (1 + <«)«« 


d-{l-\-t')x'  = 


2(1 + n- 


Then,  integrating  with  respect  to  t, 

'"     dt 


Hjnce 


k'  = 


-{-f 


=  TT, 


/e    "^dx  =  Vic, 
<x> 


BECTIFICATION  OF  CURVES. 


285 


CHAPTER  VIII. 

RECTIFICATION  AND  QUADRATURE. 

177.  The  Rectification  of  Curves.    In  the  older  geometry 
to  rectify  a  curve  meant  to  find  a  straight  line  equal  to  it  in 
length.     In  modern  geometry  it  means  to 
find  an  algebraic  expression  for  any  part  of 
its  length. 

Let  us  put  s  for  the  length  of  the  curve 
from  an  arbitrary  fixed  point  (7  to  a  vari- 
able point  P.  If  P'  be  another  position 
of  the  variable  point,  we  shall  then  have 


Fia.  50. 


As  =  PP'. 


If  PP'  becomes  infinitesimal,  it  has  already  been  shown 
(§  79)  that  we  have,  in  rectangular  co-ordinates, 


ds  =  4WTW  =  /l  +  i^£jdx  =  /l  +  [^)dy,     (1) 
and,  in  polar  co-ordinates, 

If  both  co-ordinates,  x  and  y,  are  expressed  in  terms  of  a 
third  variable  w,  we  have 


ds  =  |/( 


+ 


(ciyy 


dn  )       \du ) 
The- length  of  any  part  of  the  curve  is  then  expressed  by 


1 


f '/ 


iff) 


;i  lii'i- 


286 


THE  INTEGRAL  CALCULUS. 


the  integral  of  any  of  these  expressions  taken  between  the 
proper  limits.     Thus  we  have 


or 


(3) 


In  order  to  effect  tlie  integration  it  is  necessary  that  the 
second  members  of  (3)  shall  be  so  reduced  as  to  contain  no 
other  variable  than  that  whose  differencial  is  written;  that  is, 
we  must  have 

ds  =  f{x)(lx;    f(y)dy;    f(6)(Wy    or    f{u)du. 

Then  we  take  for  the  limits  of  integration  the  values  of 
X,  y,  6  or  u,  which  correspond  to  the  ends  of  the  curve. 

178.  Rectification  of  the  Parabola.    From  the  equation 

of  the  parabola 

y*  =  2px 

we  derive  ydy  =  pdx. 

We  shall  have  the  simplest  integration  by  taking  y  as  the 
independent  variable.     We  then  have 


</*  =  1 1  +(g'  }  'dy,   pd3  =  {p'  +  y'l'dy. 
The  formula  (C)  of  §  145  gives 

f(p'  +  ffdy  =  iy(p'  +  .V')'  +  iP'f-r^^^ 


(a) 


(p'  +  y'f 

The  method  of  §  132  gives 

{p  +  yy 

=  h-\og  p  Jr  log  ((;?'  +  2^')*  +  i/). 


A 


RECTIFICATION  OF  CURVES. 


2S7 


(3) 


(a) 


Thus,  putting  h'  =  \p  {h  —  hgp),  the  indefinite  integral 
of  (a)  is 

s  =  h'  +  i|(^'  +  f)^  +  iP  log  {{p^  +  f)^  +  I/). 

The  arbitrary  constant  //  must  bo  so  taken  tliat  s  sliall 
vanish  at  the  initial  point  of  the  parabolic  arc.  If  we  tako 
the  vertex  as  this  point,  we  must  have  s  =  Oiory  —  0.     Then 

7/  -  —^p  log  p. 

We  therefore  have,  for  the  length  of  a  parabolic  arc  from 
the  vertex  to  the  point  whose  ordinate  is  y, 


«  =  lj(p'  +  S')*  +  iP  log 


>\> 


(p'  +  .vT  +  .v 


(*) 


179.  Rectification  of  che  Ellipse.  The  formulse  for  rec- 
tifying the  ellipse  take  the  simplest  form  when  we  express  the 
co-ordinates  in  terms  of  the  eccentric  angle  u;  then  (Analyt. 
Geom.) 

x  =  acoB  u;    y  =  b  sin  u. 

We  then  have 

dx  —  —  a  sin  udw,    dy  —  h  cos  udu. 
Then  if  e  is  the  eccentricity,  so  that  a'e'  =  a'  —  J', 
ds  —  (a'  sin'  u  -J-  &'  cos'  \ifdu  =  «(!  —  e'  cos'  ufdu\ 


s  =  rt/(l—  e'  cos'  ufdu. 


This  expression  can  be  reduced  to  an  elliptic  integral:  a 
kind  of  function  which  belongs  to  a  more  advanced  stage  of 
the  calculus  than  that  on  which  we  are  now  engaged. 

It  may,  however,  be  approximately  integrated  by  develop- 
ment in  series.     We  have,  by  the  binomial  theorem, 

1  1*1 

(1  —  e'  cos'  u)^  =  1  —  ;r6'  cos'  u  —  ;z—.e*  cos*  u 


2 


2-4 

1-1-3 

2-4-G 


e  cos  u 


etc. 


288 


THE  INTEQUAL  CALCULUS. 


Ill 


i 


H 

II 


1     HI    9.^    » > 

,-■'■'  : 

The  terms  in  the  second  member  may  he  separately  in- 
tegi'ated  by  the  formulae  (0),  §  140,  by  putting  w  =  0  and 
7i  =  2,  4,  C,  etc.     We  thus  find 

2  /  cos'  ndu  =  sin  n  cos  u  •\-  w, 

4  /  cos*  luht  =  sin  ?«(cos'  ti  -\- 1  cos  n)  -f  fw; 
etc.  etc.  etc. 

Since  at  one  end  of  the  major  axis  we  have  u  =  0  and  at 
the  other  end  u  =  rr,  we  find  the  length  of  one  half  of  the 
ellipse  by  integrating  between  the  limits  0  and  7t.  Since 
sin  u  vanishes  at  both  limits,  we  have 

/    cos'  udtt  =  ^n; 
t/o  <* 


'■^  1-3 

cos*  ud2C  =  :r— .?r; 
0  2-4 


'^      ,  1-3-6 

cos"  U  =  K--i-^7t. 

0  2  •  4  •  6 


We  thus  find  by  substitution  that  the  semi-circumference 
of  the  ellipse  may  be  developed  in  powers  of  the  eccentricity 
with  the  result 

_      /'i       1  ^         3     ,       J';5_  ,  _        \ 

180.  TIte  Cycloid.  The  co-ordinates  x  and  p  of  the  cy- 
cloid are  expressed  in  terms  of  the  angle  u  through  which 
the  generating  circle  has  moved  by  the  equations  (§  80) 

X  =  a{u  —  sin  ?^); 
y  =  a{l  —  cos  It). 
Hence 

dfs'  =  dx""  +  dy^  =  rt'f (1  -  cos  n)'  +  sin'  u\du'' 

=  2«'(1  —  cos  'ii)du^  =  ^a^  sin'  ^u.du*. 

By  extracting  the  root  and  integrating, 

5  =  ^  —  4rt  cos  ^u. 


If 

meei 
have 

and 

Tl 
one 

that 


HECTlFlCATtON  OP  CURVm. 


280 


If  we  measure  tno  arc  generated  from  the  poiut  where  it 
meets  the  axis  of  abscissa^  that  is,  where  n  =  0,  we  must 
have  s  =  0  f or  7t  =  0.     This  gives 

h  =  4a 

and  s  =  4a(l  —  cos  ^«)  =  8^  sin'  \u. 

This  gives,  for  the  entire  length  of  the  arc  generated  by 
one  revolution  of  the  generating  circle, 

s  =  Set; 

that  is,  four  times  the  diameter  of  the  generating  circle. 

181.  The  Archimedean  Spiral.  From  the  polar  equation 
of  this  spiral  (§  82)  we  find 

dr  =  add. 

Hence  ds  =  a(l  +  ff'fde. 

Then  the  indefinite  integral  is  (§  147,  Ex.  1) 

5  =  1 1  ^(1  +  ^')' 4- log  q^ +(1  +  ^^))*  [ . 

If  we  measure  from  the  origin  we  must  determine  the  value 
of  0  by  the  condition  that  when  ^  =  0,  then  s  =  0.  This 
gives  log  C  =0;  .' .  C  =1. 

If  instead  of  6  we  express  the  length  in  terms  of  r,  the 
radius  vector  of  the  terminal  point  of  the  arc,  we  shall  have 

,  =  ^  -{a-  +  r-y  +  ^  log  5^ -' . 


183.    The  Logarithmic  Spiral.    The  equation  of  this 
spiral  (§  83)  gives 


Hence 


--  =  ale^^  =  Ir. 
au 

dfs  =  (1  +  rfrdS. 


To  integrate  this  differential  with  respect  to  6  we  should 

first  substitute  for  r  its  value  in  terms  of  6.    But  it  will  be 
19 


290 


THE  INTEGRAL  CALCULUS. 


better  to  adopt  the  inverse  course,  and  express  dd  in  terms  of 
dr.     We  thus  have 

ds  = ; ar; 


I 


whence 


_  (1  +  ly 


I 


r  +  5, 


0* 


s^  being  the  value  of  s  for  the  pole. 

If  we  put  y  for  the  constant  angle  between  the  radius 
vector  and  the  tangent,  then  (§§  90-92)  /=cot  y,  and  we  hg.ve 

Between  any  two  points  of  the  curve  whose  radii-vectors 
are  r„  and  i\  we  have 

s  =  (?*,  —  i\)  sec  y. 

Hence  the  length  of  an  arc  of  the  logarithmic  spiral  is  pro- 
portional  to  the  difference  betioeen  the  radii-vectors  of  the  ex- 
tremities of  the  arc. 

EXERCISE. 

1.  Show  that  the  differential  of  the  arc  of  the  lemniscato  is 


ds  = 


add 


Vl-'H  sin'  e 

(This  expression  can  be  integrated  only  by  elliptic  func- 
tions.) 

183.  The  Quadrature  of  Plane  Figures.  In  geometrical 
construction,  to  square  a  figure  means  to  find  a  square  equal 
to  it  in  area.     The  operation  of  squaring  is  called  quadrature. 

In  analysis,  quadrature  means  the  formation  of  an  algebraic 
expression  for  the  area  of  a  surface. 

In  order  to  determine  an  area  algebraically,  the  equation 
of  the  curve  which  bounds  it  must  be  given.  Moreover,  in 
order  that  the  area  may  be  completely  determined  by  the 
bounding  line,  the  latter  must  be  a  closed  curve. 

Then  whatever  the  form  of  this  curve,  every  straight  line 


QUADRATURE  OF  PLANE  FIGURES. 


291 


terms  of 


le  radius 
i  we  hpve 

[ii-vectors 


al  is  pro- 
of the  ex- 


iiiscato  is 


)tic  func- 

)ometrical 
are  equal 
adrature. 
L  algebraic 

equation 
reover,  in 
id  by  the 

light  line 


Fig.  51. 


must  intersect  it  an  -^ven  number  of  times.  The  simplest 
case  is  that  in  wliich  a  line^  paral- 
lel to  the  uxis  of  Y  cuts  the  bound- 
ary in  two  points.  Tlien  for 
every  value  of  x  the  equation  of 
the  curve  will  give  two  values  of  y 
corresponding  to  ordinates  termi- 
nating at  P  and  Q.  Let  these 
values  be  //,  and  y^. 

Then,  the  infinitesinitil  area  in- 
cluded between  two  ordinates  infinitely  near  each  other  will 
be 

O/i  -  yo¥^  =  ^<^- 

The  area  given  by  integrating  this  expression  will  be 

in  which  the  limits  of  integration  are  the  extreme  values  of  x 
corresponding  to  the  points  X^  and  JT,,  outside  of  which  the 
ordinate  ceases  to  cut  the  curve. 

The  same  principle  may  be  applied  by  taking  (x^  —  x^dy 
as  the  element  of  the  area.     We  then  have 


If  the  curve  is  referred  to  polar 
co-ordinates,  let  S  and  T  be  two 
neighboring  points  of  the  curve, 
and  let  us  put 

r^OS) 
r'^OT', 
^0  =  angle  SOT. 

If  wo  draw  a  chord  from  S  to  T, 
the  area  included  between  this  chord 
and  the  curve  will  be  of  the  third 
order  (§  78).     The  area  of  the  triangle  formed  by  this  chord 


Fio.  52. 


i>! 


Hi"  ^ 
J 


29r) 


Tllh:  TNTKORAL  CAICVLUB. 


and  the  radii  vectors  will  be  ^rr'  sin  A 6.  Now  let  A 6  be- 
come infinitesimal.  OS  '•  "'  then  approach  ?■  as  its  limit; 
the  ratio  of  sin  AS  to  A6  it  will  approach  unity,  and  the 
area  of  the  triangle  will  approach  that  of  the  sector.  Thus 
wo  shall  have,  for  the  dillerential  of  area, 

If  the  polo  is  within  the  area  enclosed  by  the  curve,  the 
total  area  will  bo  found  by  integrating  this  exprcs-sion  be- 
tween the  limits  O''  and  300'.     Thus  we  have,  for  the  total 


area. 


<T  =  \£'r\ie. 


184.  The  Parabola,    As  the  parabola  is  not  itself  a  closed 
curve,  it  bounds  no  area.     But  we  may  find  the  area  of  any 
segment  cut  off  by  a  double  ordinate 
MN.    The  equation  of  the  curve  gives, 
for  the  two  values  of  y. 


y,  =  +  'npx\  y,  =  -  ^2j)x. 

Hence 


do-  =  VSp.z^dx. 
The  indefinite  integral  is 

For  the  area  from  the  vertex  to  MN  we  put  a;,  =  OX,  and 
take  the  integral  between  the  limits  0  and  x.  Calling  this 
area  c,,  we  have 


cr,  =  f  V2px^  .X,  =  ^x^y^  =  K X2y,. 

Because  2i/,  =  MN,  it  follows  that  the  areaABMN=2x^y^. 
Hence: 

Theorem.  The  area  of  a  parabolic  segment  is  two  thirds 
that  of  its  circumscribed  rectangle. 


18d 

curve, 

consid< 

Let 

curve. 


QUADIiATl/IihJ  OF  PLANE  FlGUUEa. 


21)13 


k  Ad  be- 
ts limit; 
and  tlio 
r.     Thus 


nrve,  the 
isiun  bo- 
the  total 


If  a  closed 
3a  of  any 


M 


185.  The  Circle  and  the  EUipsc.     Referring  the  circle 
of  radius  a  to  the  centre  as  the  origin,  the  values  of  y  will  bo 


«\i 


y=±  («'  -  xy. 


Hence 


/(y.  -  yo)^/^  =  ''iy*  («'  -  ^')*^^^ 


.X 


=  x(d'  -xy-{-  a'  sin  <-  »>  -  4-  h. 

This  expression,  taken  between  appropriate  limits,  will 
give  the  area  of  any  portion  of  the  circle  contained  between 
two  ordinates. 

Taking  the  integral  between  the  limits  —  a  and  -|-  «  gives, 
for  the  area  of  the  circle, 

0-  =  «»  8in<-«  (+  1)  -  rt'  sin(-«  (-  1)  =  na\ 

The  Elli])se.  From  the  equation  of  the  ellipse  referred  to 
its  centre  and  axes,  namely. 


\  OX,  and 
dling  this 


•wo  thirds 


we  find 
The  entire  area  will  be 

,  +  a 


V  =  ±  -  Vrt'  -  x\ 


+  a 


(Vi  -  y.¥^  =  ^-        v^'  -  ^f^^  =  ^«*- 

■  a  '■x/—a 

The  last  integration  is  performed  exactly  as  in  the  case  of 
the  circle. 

186.  The  Hyperbola.  Since  the  hyperbola  is  not  a  closed 
curve,  it  does  not  by  itself  enclose  any  area.  But  we  may 
consider  any  area  enclosed  by  an  hyperbola  and  straight  lines. 

Let  us  first  consider  the  area  A  PM  contained  between  the 
curve,  the  ordinate  MP,  and  the  segment  AM  of  the  major 


2U4 


TUK  l^TEURAL  CALCULUS. 


Ifl 


axis.     Tho  oquivtion  of  the  hyperbolu  referred  to  its  centre 
autl   axes  gives,  for  the  value  of 
ij  ill  terms  of  x, 

h 


y=z 


a 


S/x? 


a\ 


Fio.  64. 


If  wo  put  a:,  for  the  vahio  of 
the  abscissa  0  M,  tlicn,  since 
OA  =  a,  tlie  area  AMI*  will  bo 
equal  to  the  integral 

/(.•-a')«^  =  l.(.'-„-)»-|'l„gg+g-l)']; 
and  for  tho  definite  integral  between  the  limits  a  and  x, 
Area.iP^=l|(..-.').-flo«g  +  g-l)'] 

=  |^y-flogg  +  g-l)*]. 

Now,  \xy  is  the  area  of  the  triangle  OPM;  we  therefore 
conclude  that  the  second  term  of  the  expression  is  the  area 
included  between  OA,  OP  and  the  hyperbolic  arc  A  P. 

Much  simpler  is  the  area  included  between  the  curve,  one 
asymptote,  and  two  parallels 
to  the  other  asymptote.  The 
equation  of  the  hyperbola  re- 
ferred to  its  asymptotes  as 
axes  of  co-ordinates  (which 
axes  are  oblique  unless  the 
hyperbola  is  equilateral)  may 
be  reduced  to  the  form  fio.  55. 


xy 


ah 
2  sin  a' 


a  CI 

out] 

i» 

y 

M 

X 

n 


I  X, 


-)'] 


therefore 
the  area 
P. 
urve,  one 


QUADRATUllE  OF  PLA^E  FIUUliES. 


296 


a  being  the  angle  between  thu  axes.  We  readily  see  that  the 
difforential  of  the  area  is  ydx  x  sin  a  instead  of  ydx  simply. 
Hence  for  the  area  we  have 


ab 


ab 


/J         />ao ,        no , 
y  sin  adx  —  I  -  ax  =  --  log  ex. 


If  we  take  the  area  between  the  limits  OM=x^  and  OM 

''» ab  ,        ab 


x^,  the  result  will  bo 


J/»-*^»  ao  J    ^(to  .      a;, 


We  note  that  this  area  becomes  infinite  when  x^  becomes 
zero  or  when  x^  becomes  infinite,  showing  that  the  entire  area 
is  infinite. 

187.  The  Lemniscatc.  The  equation  of  this  ourve  in 
polar  co-ordinates  is  (§  81) 

?•'  =  a'  cos  3^. 

It  will  bo  noted  that  r  becomt^  imaginary  when  6  is  con- 
tained between  45°  and  135°,  or  between  225"  and  315°. 
The  integral  expression  for  the  area  is 

^fr'de  =  ia'f  COS  20de  =  ia'  sin  26. 

To  find  the  area  of  the  right-hand  loop  of  the  curve  we 
must  take  this  integral  between  the  limits  6  =  —  45°  and  0  = 
+  45°,  for  which  sin  26/  =  -  1  and  +  1.     Hence 

Half  area  =  !«'; 
Total  area  =  «'. 

Hence  the  area  of  each  loop  of  the  lemniscate  is  half  the  square 
on  the  semi-axis. 

188.  The  Cycloid.  By  differentiating  the  expression  for 
the  abscissa  of  a  point  of  the  cycloid  we  have 

dx  =z  a(l  —  cos  u)du. 
Hence 


I 


l;^  3 


y,H 


1 J     mwM 


m 


296 


TUB  INTEGRAL  CALCULUS. 


/  ydx  =a^  I  (1—  cos  uYdu—a'^  I  (| — 2  cos  w  +  i cos 2u)du. 

The  indefinite  integral  is 

|?A  —  2  sin  w  +  i  sin  2i/. 

To  find  the  whole  area  wc  take  the  definite  integral  between 
the  limits  0  and  27r,    Thus  we  find 

Area  of  cycloid  =  3;ra', 

or  three  times  the  area  of  the  generating  circle. 

EXERCISES. 

I.  Show  that  the  theorem  of  §  184  is  true  only  of  the  pa- 
rabola. 

To  do  this  we  must  find  what  the  equation  of  a  curve  must  be  in  order 
that  the  theorem  may  be  true.    The  theorem  is 


/ 


ydx  =  Ixy. 

Differentiating  both  members,  we  have 

ydx  =  Ixdy  +  lydx ;  , 

•    0  =  ^ 

'  '    y      x' 

Then,  integrating  both  members, 

log y"^  =  \ogcx',    .' .  y"^  =  ex, 

c  being  an  arbitrary  constant.    This  is  the  equation  of  a  parabola  whose 
parameter  is  ^c. 

2.  Show  that  the  equation  of  a  curve  the  ratio  of  whose 

area  to  that  of  the  circumscribed  rectangle  is  m  :  n  must  be 

of  the  form 

y^  =  cx^~^. 


CUBATURE  OF  VOLUMES, 


297 


OS  %u)du. 


I  between 


f  the  pa- 
be  in  order 


abola  whose 

>  of  whose 
I  must  be 


CHAPTER  IX. 

THE  CUBATURE  OF  VOLUMES. 

189.  General  Formulce  for'  Cubature.  In  the  ancient 
Geometry  to  C2cbe  a  solid  meant  to  find  the  edge  of  a  cube 
whose  volume  should  be  equal  to  that  of  the  solid.  In  Ana- 
lytic Geometry  it  means  to  find  an  expression  for  the  volume 
of  a  solid. 

Let  us  have  a  solid  the  bounding  surface  of  which  is  de- 
fined by  an  equation  between  rectangular  co-ordinates.     Let 
the   solid    be  cut  by  a 
plane  PL  parallel  to  the 
plane  of  YZ,  and  let  u 
be  the  area  of  the  plane 
section  thus  formed.     If 
we  now  cut  the  solid  by 
a  second  plane,  parallel 
to  PL  and  infinitely  near 
it,  that  portion  of  the 
solid  contained  between 
the  planes  will  be  a  slice  of  area  u  and  thickness  dx,  dx  being 
the  infinitesimal  distance  between  the  planes. 

If,  then,  we  put  v  for  the  volume  of  that  part  of  the  solid 
contained  between  any  two  planes  parallel  to  YZ,  we  have 


Fig.  5G. 


and 


dv  =  udx, 
V  —   I     udx, 

t/Xa 


(1) 


x^  and  x^  being  the  distances  of  the  catting  planes  from  the 
origin  0, 


298 


TUB  INTEGRAL  CALCULUS. 


m 


i'i 


r  'H 


i«  1 


t 

I 


If  we  take  for  x^  and  x^  the  extreme  values  of  x  for  any 
part  of  the  solid,  the  above  expression  will  give  the  total  vol- 
ume of  the  solid. 

In  order  to  integrate  (1),  we  must  express  t*  as  a  function 
of  X.  That  is,  we  must  find  a  general  expression  in  terms  of 
X  for  the  area  of  any  section  of  the  solid  by  a  plane  parallel 
to  that  of  XY.  This  is  to  be  done  by  the  equation  of  the 
bounding  surface  of  the  solid. 

Of  course  we  may  form  the  infinitesimal  slices  by  planes 
perpendicular  to  the  axis  of  I^  or  of  Z  as  well  as  of  X. 

190.  The  Sphere.  The  equation  of  a  sphere  referred  to 
its  centre  as  the  origin  is 

If  we  cut  the  sphere  by  a  plane 
FMQ  parallel  to  the  plane  of 
YZ,  and  having  the  abscissa  OM 
=  X,  the  equation  of  the  circle  of 
intersection  will  be 

?/"  ^  z'  =  a'  —  x'; 

that  is,  the  radius  MP  of  the  circle  will  be  Va"*  —  x\  and  its 
area  will  be  7r{a'  -  x^).  Hence  the  differential  of  the  vol- 
ume of  the  sphere  will  be 

dv  =  7t(a^  —  x^)dx, 
and  the  indefinite  integral  will  be 

V  =  n{a^x  —  \x^)  -f  C. 
The  extreme  limits  of  x  for  the  sphere  are 
x^~:  —a    and    x^-=  -\-  a. 
Taking  the  integral  between  these  limits,  we  have 
Volume  of  sphere  =■  |;ra', 


Fig.  57. 


X 


CtJBATURE  OF  VOLUMES. 


■  for  any 
total  vol- 

function 

terms  of 

I  parallel 

n  of  the 

by  planes 


'299 


',  and  its 
the  vol- 


191.  Volume  of  Pyramid.  Let  the  pyramid  bo  placed 
with  its  vertex  at  the  ori-' 
gin,  and  its  base  parallel  to 
the  plane  of  XY,  Let  us 
also  put  h  =  OZ  its  alti- 
'ude;  a,  the  area  of  its  base. 
Let  it  be  cut  by  a  plane 
EFGH  parallel  to  its  base. 

It  is  shown  in  Geometry 
that  the  section  EFGH  is  ^ 
similar  to  the  base,  and  that 
the  ratio  of  any  two  homologous  sides,  as  EFtm^AB,  is  the 
same  as  the  ratio  OL  :  OZ.  Because  the  areas  of  polygons 
are  proportional  to  the  squares  of  their  homologous  sides, 

.-.Area  EFGH  :  Area  ABCD  =:  OIJ  :  0Z\ 
Putting  Area  ABCB  =  a,  OL  =  z  and  OZ  =  h, 


Fig.  58. 


ArGB,EFGII  = 


az 


The  volume  of  the  pyramid  is  therefore 


'^nz'dz      1    , 

7?-  =  a"*- 


That  is,  one  third  the  altitude  into  the  base. 
The  same  formula}  apply  to  the  cone. 

193.  Tr'e  Ellipsoid.    The  equation  of  the   ellipsoid  re- 
ferred to  its  centre  and  axes  is 


+  -  =  1 


a,  b  and  c  being  the  principal  semi-axes. 

If  we  cut  the  ellipsoid  by  the  plane  whose  equation  is 
X  =  a;',  the  equation  of  the  section  will  be 

0        c  a 


300 


THE  INTEGRAL  CALCULUS. 


i 


)i' 


(:1 


This  is  the  equation  of  an  ellipse  whose  semi-axes  are 


a 


Va'  - 


X 


/a 


and     -  Va^  — 


X 


n 


a 


Hence  its  area  is 


7tlc{a'  -  x") 


a 


Fio.  59. 


Then,  by  integration  between  the  limits  —a  and  -{-a,  we  find 
Volume  of  ellipsoid  =  ^mihc. 

From  the  known  expression  for  the  area  of  an  ellipse  (Ttah) 
it  is  readily  found  that  the  volume  of  an  elliptic  cylinder  cir- 
cumscribing any  ellipsoid  is  27tabc.     Hence  we  conclude: 

2^ he  volutiie  of  an  ellipsoid  is  two  thirds  that  of  any  rigJit 
elliptic  cylinder  circumscribed  about  it. 

193.  Volume  of  any  Solid  of  devolution.  In  erder  that 
a  solid  of  revolution  may  have  a  well-defined  volume  it  must 
be  generated  by  the  revo- 
lution of  a  curve  or  un- 
broken series  of  straight 
or  curve  lines  terminating 
at  two  points,  Q  and  M, 
of  the  axis  of  revolution. 

As  an  element  of  the  volume  we  take  two  planes  infinitely 
near  each  other  and  perpendicular  to  the  axis  of  revolution. 
Every  such  plane  cuts  the  solid  in  a  circle.  If  we  place  the 
origin  at  0,  take  the  axis  of  revolution  as  that  of  X,  and  let 

OM  =  xhe  the  abscissa  of  any  point  P  of  the  curve,  and 

MP  =  y  its  ordinate, 
then  the  section  of  the  solid  through  M  will  be  a  circle  of  ra- 
dius ?/,  whose  area  will  therefore  be  Try^. 

Hence  the  volume  contained  between  two  planes  at  distance 

dx  will  be 

Tty^dx, 

and  the  volume  between  two  sections  whose  abscissas  are  x„ 

and  X.  will  be 

V=    I   'tthMx.  (1) 


/    TTt/^dx. 


iti|:t 


CUBATUBE  OF  VOLUMES. 


301 


,re 


,  we  find 

se  (Ttah) 
ider  cir- 
ide: 
^ly  rigJit 

ier  that 
it  must 


|R 

ifinitely 
Dlution. 
ace  the 
md  let 
,  and 

3  of  ra- 

listanco 

are  x„ 

(1) 


If  the  two  co-ordinates  are  expressed  in  terms  of  a  third 
variable  w  by  the  equation*- 


we  have 


dx  =■  ff)'{ii)du. 


Putting  u^  and  «,  for  the  values  of  u  corresponding  to  x^ 
and  a;,,  the  expression  (1)  for  the  volume  will  become. 


V=7r  I     [>p{u)Y(p'(7i)du. 

tJuo 


(2) 


The  equations  (1)  and  (2)  give  the  volume  AA'B'B  gen- 
erated by  the  revolution 
of  any  arc  AB  of  the 
given  curve,  and  of  the 
ordinates  MA  and  NB 
of  the  extremities  of  the 
arc.  The  limits  of  in- 
tegration for  X  are  OM 
=  x„  and  OW  =  x^.  To 
find  the  entire  volume  generated  we  must  extend  these  limits 
to  the  points  (if  any)  at  which  the  curve  intersects  the  axis  of 
revolution. 

104.   The  Paraboloid  of  Revolution.    The  equation  of  the 
parabola  being  y'  =  ^px,  we  readily 
find  from  (1)  a  result  leading  to  the 
following  theorem,  which  the  student 
should  prove  for  himself: 

Theorem.  The  volume  of  a  para- 
boloid  of  revolution  is  one  half  that 
of  the  circumscribed  cylinder. 

195.  The  Volume  Generated  by 
the  Revolution  of  a  Cycloid  around 
Us  Base,     From  the  equations  of  the  cycloid  in  terms  of 


FiQ.  61. 


302 


TBE  INTEGRAL  CALCULUS. 


lii 


i*j 


:! 


i 


1' 


i  , 

* 

i'l' 

1- 

!'1 

r    i 

ink 

1 

the  ajgle  through  which  the  generating  circle  has  moved, 
we  find  the  element  of  the  volume  to  be 


dV=  7ra\l  —  cos  ttydu. 


fience 


V  =  Tca'  /  (1  —  3  coa  ?*  +  3  cos'  u  —  cos'  ii)dti. 

By  the  method  of  §§  149,  150,  with  simple  reductions,  we 
find 

/  cos'  udu  —  \u  -f-  \  sin  3?*; 

/  cos'  udv>  =  /  (1  —  sin'  u)d.mi  w  =  sin  w  —  ^^  sin'  u 

=  I  sin  w  -j-  ^  sin  3w. 
We  thus  find,  for  the  indefinite  integral, 

V  —  na^{^\'i(,  —  Y  sin  w  + 1  sin  %u  —  -^  sin  3w). 

The  total  volume  formed  by  the  revolution  of  one  arc  of 
the  cycloid  is  found  by  taking  the  integral  between  the  limits 
w  =  0  and  %i  =  2;r.     The  volume  thus  becomes 

F==5;r'«', 

from  which  follows  the  theorem: 

The  volume  generated  hy  the  revolution  of  a  cycloid  around 
its  base  is  five  eighths  that  of  the  circumscrihed  cylinder. 

196.  The  Hyperboloid  of  Revolution  of  Two  Nappes. 
This  figure  is  formed  by  the  revolution  of  an  hyperbola  about 
its  transverse  axis.  The  general  expression  for  the  volume  is 
found  to  be 

F=g(a;'-3ri'.r  +  ;0, 

h  being  the  arbitrary  constant  of  integration.  If  we  consider 
that  part  of  the  infinite  solid  cut  off  by  a  plane  perpendicular 
to  the  transverse  axis,  we  must  determine  h  by  the  condition 


tha 

be 

wil 


CUBATURE  OF  VOLUMES. 


303 


that  V  shall  vanish  when  x  =  a,  because  then  the  plane  will 
be  a  tangent  at  the  vertex  o'  the  hyperboloid,  and  the  volume 
will  become  zero.     This  condition  gives 


h  =  3a'  -  a'  =  2a\ 


Thus  wo  have 


V  =  —(x^  -  Sa'x  +  2a')  =  -^,(x  -  ay(x  +  2a).     (1) 

By  the  same  revolution  whereby  the  hyperbola  describes  an 
hyperboloid  of  revolution  the  asymptotes  will  describe  a  cone. 
Let  us  compare  the  volume  just  found  for  the  hyperboloid 
with  that  of  the  asymptotic  cone,  cut  off  by  the  same  plane 
which  cuts  off  the  hyperboloid.  The  equation  of  the  generat- 
ing asymptote  being 

ay  =  bx, 


we  find  for  the  volume  of  the  cone 


F'  = 


TT 


nV 


P^'-'M' 


(^^) 


The  difference  between  (1)  and  (2)  will  be  the  volume  of 
the  cup-shaped  solid  formed  by  cutting  the  hyperboloid  out 
of  the  cone.     Calling  this  volume  F",  we  find 


V"  =  7rb\x  -  ^a). 


(3) 


This  is  equal  to  the  volume  of  a  circular  cylinder  of  which 
the  diameter  is  the  conjugate  axis  of  the  hyperbola,  and  the 
altitude  x  —  fa. 

This  result  is  intimately  associated  with  the  following 
theorem,  the  proof  of  which  is  quite  easy: 

If  a  plane  peiyendicular  to  the  axis  of  revolution  cut  an 
hyperbola  of  two  nappes  and  its  asymptotic  cone,  the  area  of 
the  plane  contained  between  the  circular  sections  is  constant 
and  equal  to  the  area  of  the  circle  whose  diameter  is  the  con- 
jugate axis. 


f 


304 


THE  INTEGRAL  CALCULUS. 


My 


m 


:i 


V  i 


<  i 


<l 

p 

P 

O 

M 

X 

197.  Rinfj-shaped  Solids  of  RevoluiiGn.  If  any  com- 
pletely bounded  plane  figure  APQB  revolye  around  an  axis 
OX  lying  in  its  own  plane,  but 
wholly  outside  of  it,  it  will  describe 
a  ring-shaped  solid. 

To  inyostigate  such  a  solid,  let  the 
ordinate  MP  cut  the  figure  in  the 
points  Q  and  P,  and  let  us  put 

yr  =  ^Q>      y,  =  ^P-  Fig.  62. 

The  points  P  and  Q  will  describe  two  circles  which  will 
contain  between  them  the  sectional  area 

Taking  two  ordinates  at  the  infinitesimal  distance  dx,  the 
corresponding  infinitesimal  element  of  volume  will  be 

dV=7t{y-^-y,yh'.  (1) 

The  integral 

V=7t  r\y,'  -  y,')dx  =  Tt  r\y,  +  y^)  (y,  -  y)dz 

will  express  the  volume  of  that  part  of  the  solid  contained  be- 
tween the  two  planes  whose  respective  abscissas  are  x^  and  x^. 
By  taking  for  x^  and  x^  the  abscissas  of  the  extreme  points 
A  and  B,  V  will  express  the  total  volume  of  the  solid. 

198.  Application  to  the  Circular  Ring.  Let  the  figure 
AB\)Q9>  circle  of  radius  c,  whose  centre  is  at  the  distance  h 
from  the  axis  of  revolution.     Let  us  also  put 

a  =  the  abscissa  of  the  centre. 

We  then  have 

y,  =  h-^Vc'-{x~aY', 
y,  +  2/,  =  2^*; 

y.-y.  =  ^  ^^"-  (^  -  «r; 


w 


CUBATURE  OF  VOLUMES. 


305 


any  com- 
i  an  axis 


y 


Rrhich  will 


ce  dx,  the 
be 

(1) 


-  y)dz 

itained  be- 
x^  and  ajj. 
me  points 
[id. 

the  figure 
distance  h 


V=  inh  r'[c'  -{x-  ayfdx. 

The  limits  of  integration  for  the  whole  volume  are 

x^  =  a  —  c    and    x^  =  a  -\-  c. 
If  we  put 


z  =  x 
the  total  volume  will  become 


a, 


V=^7tl)f_y{c'  -Z')\lz. 


By  substituting  the  known  value  of  the  definite  integral, 
we  have 

V='Zn'bc\ 

The  area  of  the  generating  circle  is  Ttc^,  and  the  circumfer- 
ence of  the  circle  described  by  its  centre  is  2;rZ>.  The  product 
of  these  two  quantities  is  %7t%c^.     Hence: 

The  volume  of  a  circular  ring  is  equal  to  the  product  of 
the  area  of  its  cross-section  into  the  circumference  of  its  central 
circle. 

EXAMPLES  AND  EXERCISES. 

1.  Compare  the  cycloid  with  the  semi-ellipse  having  the 
same  axes  as  the  cycloid,  and  show  the  following  relations  be- 
tween them: 

a.  The  maximum  radius  of  curvature  of  the  ellipse  (at  the 
point  B)  is  greater  than  that  of  the  cycloid  in  the  ratio 
Ti"^  :  8,  or  5  :  4,  nearly. 

/?.  The  area  of  the  semi-ellipse  is  greater  than  that  of  the 
cycloid  in  the  ratio  tt  :  3. 

y.  The  volume  of  the  ellipsoid  of  revolution  around  the 
axis  OX  is  greater  than  that  generated  by  the  revolution  of 
the  cycloid  iu  the  ratio  16  :  15. 
20 


i! 


m 


It; •If!  * 


!'• 


li. 


4:f 


306 


THE  INTFAHtAL  CALCULUS. 


Q. 


o 


4v 


Fio.  63. 


''r' 


190.  Quadrahirc  of  Surfaces  of 
Revolution,     Let  us  put 

^5  E  a  small  arc  PQ  oi  a  curve  re- 
volving round  an  axis  0X\ 
y  E  the  distance  of   /*  from  the ' 
axis  0X\ 

y'  =  the  distance  of   Q  from   the 
axis  OX, 

Considering  Js  as  a  straight  line, 
the  surface  generated  by  it  will  bo  the  curved  surface  of  the 
frustum  of  a  cone.     If  we  put 

J(T  =  the  area  of  this  curved  surface,  we  have,  by  Geometry, 

Jo-  =  7r{y  -f  y')^s. 

Now  let  J.<?  become  infinitesimal.  Then  y'  will  approach  y 
as  its  limit,  and  we  shall  have,  for  the  differential  of  the  sur- 
face. 


d(T  =  ^nyils  =  27ry\  1  -j- 


\dxl 


dx. 


This  expression,  when  integrated  between  the  limits  x^  and 
iCj,  will  give  the  area  of  that  portion  qf  the  surface  for  which 
the  co-ordinates  x  are  contained  betv/een  x^  and  .r,. 

The  modifications  and  transformations  of  this  formula  eo 
as  to  apply  it  to  cases  when  another  axis  than  that  of  Y  is 
the  axiH  of  revolution,  or  when  the  equation  of  the  curve  is 
not  in  the  form  y  =  0(.r),  can  be  made  by  the  student  himself. 

300.  Examples  of  Surfaces  of  Revolution.  The  process 
of  applying  the  general  formula  for  dcr  to  special  cases  is  so 
like  that  already  followed  in  quadrature  and  cubature  that 
the  briefest  indications  will  suffice  to  guide  the  student. 

Surface  of  the  Siiliere.  Supposing  the  equation  of  the  gen- 
erating circle  to  be  written  in  the  form 

^'  +  y'  =  ^«% 


SURFACES  OF  REVOLUTION. 


307 


le  of  tlie 


lometry. 


)roach  y 
the  sur- 


;s  x^  and 
T  which 

tnula  eo 
of  Fis 
curve  is 
himself. 

process 
ses  is  so 
ire  that 

Qt. 

the  geii- 


wo  shall  find  the  differential  of  the  surface  to  bo 

d(T  =  27railx. 

From  this  we  may  easily  prove  the  fo. lowing  : 

TuEOREM  I.  If  a  sphere  be  cut  by  any  number  of  parallel 
and  equidistant  ])lanes,  the  curved  surfaces  (f  the  S2)herical 
zones  contained  between  the  planes  will  all  be  equal  to  each 
other. 

Theorem  II.  The  total  surface  of  a  sphere  is  equal  to  the 
product  of  its  diameter  and  circumfei'ence. 

Surf  ace  generated  by  the  Revolution  of  a  Cycloid.  Wo  shall 
find  the  differential  of  the  surface  to  bo 

d(T  =  ^Tta"^  sin^  ^udu. 

By  a  formula  found  in  Trigonometry,  we  have 

8  sin'  V  —  G  sin  V  —  2  sin  3v. 

Hence,  putting  v  =  ^u, 

dcr  =  4;ra'  (3  sin  v  —  sin  dv)dv. 

The  whole  surface  is  obtained  by  integrating  between  the 
limits  u  =  0  and  u  =  27r;  that  is,  v  =  0  and  v  =  n.  We 
thus  find,  for  the  total  surface. 

Hence  the  theorem: 

The  total  surface  generated  by  the  revolution  of  a  cycloid 
about  its  base  is  four  thirds  the  surface  of  the  greatest  in- 
scribed sphere. 

The  Paraboloid  of  Revohition.  Taking  the  integral  be- 
tween the  limits  zero  and  x,  we  have  for  the  curved  surface 

THE  END. 


